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https://mathoverflow.net/questions/127284 | 5 | I am trying to find some examples of Fraïssé classes that do **not** have the strong amalgamation property. Anyone?
| https://mathoverflow.net/users/33032 | A Fraïssé class without the strong amalgamation property. | **In this edit, the relevant theorem is properly stated, and a previous "fake" example has been replaced by a genuine one (in light of corrective comments by Joel Hamkins and Emil Jeřábek).**
Goldstern and Hamkins have provided nice examples. I will add:
**Theorem** *A Fraïssé class has the strong amalgamation prop... | 11 | https://mathoverflow.net/users/9269 | 127297 | 70,968 |
https://mathoverflow.net/questions/127142 | 18 | It is a remarkable property of uncountable compact metric spaces that each of them contains a homeomorphic copy of the Cantor set. In general, one cannot expect containment of Cantor cubes (in particular, in the case of scattered compact spaces) but instead different totally disconnected subspaces occur quite often. My... | https://mathoverflow.net/users/31994 | Closed totally disconnected subspaces | EDIT: I tried to improve the presentation. I hope that it is a bit more readable now.
I think that the following construction gives a consistent counter-example. It uses the combinatorial principle $\diamondsuit$, which implies CH (the continuum hypothesis) and is independant from ZFC.
I'll work with ordinals, and... | 7 | https://mathoverflow.net/users/29491 | 127300 | 70,971 |
https://mathoverflow.net/questions/127291 | 4 | Let $E$ be a rank $r$ bundle on a quasiprojective scheme. There is a natural map
$$
\Lambda^r H^0(E) \to H^0(\Lambda^r E)
$$
It does not have to be surjective as can be seen from the case when $E$ is a direct sum of line bundles. However, is it true that $E$ can always be twisted by an ample line bundle so that this... | https://mathoverflow.net/users/11051 | Sections of determinant bundle | Assuming $H^0(X,\mathcal O\_X)$ is a field $K$, yes. We will prove a slightly more general fact: For all pairs of coherent sheaves $A$, $B$, for some ample $L$, $H^0(X,A \otimes L) \otimes H^0(X,B \otimes L) \to H^0(X,A \otimes B \otimes L^2)$ is surjective.
This is more general because it is sufficient to show $\oti... | 10 | https://mathoverflow.net/users/18060 | 127305 | 70,974 |
https://mathoverflow.net/questions/127275 | 15 | I am not quite sure that this question is appropriate for Mathoverflow, yet I would be deeply grateful for any hint: what happens in section 3 of Beilinson A., Bernstein J., Deligne P., Faisceaux pervers//Asterisque 100, 1982, 5-171? Is there any statement that is important for the following sections of this treatise? ... | https://mathoverflow.net/users/2191 | What is the purpose of section 3 of BBD? | I'm not an expert at BBD, but the introduction explains that chapter 3 is supplementary technical information that can be skipped. For example, if you want to work with $\mathbb{Z}$-coefficients, or the filtered derived category, you might find chapter 3 useful. As far as a "plan" of BBD is concerned, the introduction ... | 7 | https://mathoverflow.net/users/121 | 127307 | 70,975 |
https://mathoverflow.net/questions/127313 | 2 | Is there a good way to see what the units of the group ring $\mathbb{F}\_p[\mathbb{F}\_p]$ (p is a prime) are?
| https://mathoverflow.net/users/27253 | Description of the units of the group ring Fp[Fp] ? | You can use that $\mathbb{F}\_p[\mathbb{Z}/p]\cong \mathbb{F}[\epsilon]/\epsilon^p,$ where $\epsilon=1-\sigma$ for $\sigma$ a generator of the group $\mathbb{Z}/p$. This is an Artin local ring, and from this you can get in any number of ways that units are $R\setminus I$ where $R$ is your ring and $I$ is the ideal span... | 5 | https://mathoverflow.net/users/7108 | 127315 | 70,979 |
https://mathoverflow.net/questions/127268 | 13 | Let $(M,\omega)$ be a compact symplectic manifold and $(L,\nabla)$ a prequantum line bundle. There are two schemes to quantize this data:
1. Choose a polarization $P$ of $M$ and define the quantum Hilbert space to be sections of $L$ that are parallel along $P$. This space admits an action of the Poisson algebra of sm... | https://mathoverflow.net/users/4622 | Reconciling two notions of geometric quantization. | Klaas Landsman has been proposing that geometric quantization is best understood as taking values in [KK-cocycles](http://ncatlab.org/nlab/show/KK-theory). (A quick review of this idea is for instance around p. 134 of his student's thesis available [here](http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf).) If one re... | 5 | https://mathoverflow.net/users/381 | 127317 | 70,980 |
https://mathoverflow.net/questions/127311 | 3 | Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition:
$$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$
The series solution is $$\sum\_{n=1}^\infty \frac{4\sin((2n-1)\pi x)\mbox{sinh}((2n-1)\pi y)}{(2n-1)\pi\mbox{sinh}((2n-1)\pi)}$$
The value of $u(1... | https://mathoverflow.net/users/10583 | A series question related to solution of Laplace equation | Let $v,w,z$ be the functions obtained from $u$ by composing with a rotation of angle $\frac\pi4,\frac\pi2,\frac{3\pi}4$ about $(\frac12,\frac12)$. The sum $u+v+w+z$ is a harmonic function, whose value at the boundary is constant equal to $1$. Thus $u+v+w+z\equiv1$. This gives you $4u\left(\frac12,\frac12\right)=1$.
| 3 | https://mathoverflow.net/users/8799 | 127320 | 70,981 |
https://mathoverflow.net/questions/127289 | 5 | (1) Suppose $\pi$ is a set of primes and $G$ is a $\pi$-divisible nilpotent group, i.e., for any $g \in G$ and $p \in \pi$, there exists $x \in G$ such that $x^p = g$. Is it necessary that all the homology groups of $G$ for the trivial group action with coefficients in $\mathbb{Z}$ are also $\pi$-divisible groups? I am... | https://mathoverflow.net/users/3040 | Homology groups of divisible and powered (nilpotent) groups | In the nilpotent case - YES:
Let $\pi$ be any set of primes. A group $K$ is called $\pi$-local if for every $n$ with no prime divisors in $\pi$ the $n$th power function $k\mapsto k^n$ is a bijection $K\to K$. If $G$ is any nilpotent group then there exists a $\pi$-localization of $G$ - it is the initial homomorphism
$... | 6 | https://mathoverflow.net/users/16678 | 127330 | 70,987 |
https://mathoverflow.net/questions/114344 | 21 | I submitted a 24 pages paper to a good journal - say usually in the top 10-20 - of pure maths, and after 14 months from the submission I haven't received any report. The last news I had from the editor date last may. Then I tried to contact him in september but no answer. What would you do? Wait? Write again? Withdraw ... | https://mathoverflow.net/users/4096 | 13 months and not even one report. what would you do? | Hey one month ago I finally received a pretty positive report, asking for some small changes!
Persistence is the key ! :)
| 12 | https://mathoverflow.net/users/4096 | 127333 | 70,989 |
https://mathoverflow.net/questions/127332 | 4 | In the course of preparing lessons on projective geometry I want to give an account on the historical development. It is easy to obtain an overview of the history starting with G. Desargues. And with respect to older sources <http://en.wikipedia.org/wiki/Projective_geometry> is of great help. But what I am not sure abo... | https://mathoverflow.net/users/nan | Was Desargues more an Euclid or an Eudoxos? | I guess you'll find what you need in the monograph by J.V. Field and J.J. Gray, [The Geometrical Work of Girard Desargues](http://journals.cambridge.org/abstract_S0007087400024444).
>
> They give substantial critical and
> exegetical commentaries as well as
> valuable introductory essays placing
> Desargues and ... | 8 | https://mathoverflow.net/users/11260 | 127337 | 70,991 |
https://mathoverflow.net/questions/127335 | 0 | Imagine one generates some form of random graph (e.g. a random geometric graph) and via simulation, calculates the probability that there exists an edge-wise path between all vertices in the graph as a function of the number of vertices and/or the allowed edge lengths in an area of fixed size (e.g. a square or circle).... | https://mathoverflow.net/users/32871 | Is there a proper way to define a threshold vertex density for a random graph s.t. the graph is fully connected? | The technical term is just that: "connectivity threshold" or "threshold for connectivity".
In general, the connectivity threshold and the giant component threshold are different. Take, for example, the percolation thresholds on the path $P\_{n/2} + K\_{n/2}$. In this case, the giant component threshold is $p = 1/n$ b... | 1 | https://mathoverflow.net/users/8894 | 127339 | 70,992 |
https://mathoverflow.net/questions/127322 | 18 | Being a new member, I am not yet sure whether my question will be taken as a research level question (and thus, appropriate for MO). However, I have seen similar questions on MO, couple of which led me asking mine, and I seem to not be able to find many resources except discussion on FOM and MO. So, any references to r... | https://mathoverflow.net/users/33039 | When does $ZFC \vdash\ ' ZFC \vdash \varphi\ '$ imply $ZFC \vdash \varphi$? | $\def\zfc{\mathrm{ZFC}}\def\pr{\operatorname{Prov}\nolimits}$The statement
>
> $\zfc\vdash\pr\_\zfc(\ulcorner\varphi\urcorner)$ implies $\zfc\vdash\varphi$ for every sentence $\varphi$ in the language of $\zfc$
>
>
>
is equivalent to the statement that $\zfc$ is either inconsistent or $\Sigma^0\_1$-sound: the ... | 23 | https://mathoverflow.net/users/12705 | 127340 | 70,993 |
https://mathoverflow.net/questions/127338 | 2 | Suppose $K=\mathbb{Q}\_p$ ist the p-adic field and $G$ a finite group with a normal subgroup $S$. I would like to get all the irreducible representations of $G$ over $K$ .
I tried to solve this by using induction from the irreducible representations of $S$ and then checked with Mackey if they are irreducible or not.
... | https://mathoverflow.net/users/33046 | Representations over $\mathbb{Q}_p$ | For finite groups and a field $F$ of characteristic zero, you can identify the $G$-endomorphisms
of two induced representations
$$ Hom\_G( I\_H^G \pi, I\_K^G \sigma) $$
with the space of functions $f: H \backslash G / K \rightarrow Hom\_F(V\_\pi, V\_\sigma\*)$ with $f(hgk)= \pi(h) f(g) \sigma(k)$.
If $H=G$ and $\pi = \... | 2 | https://mathoverflow.net/users/10400 | 127342 | 70,994 |
https://mathoverflow.net/questions/127207 | 4 | Hi,
I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, flat scheme of relative dimension $n$ and $Z\subset X\_s$ be an equidimensional closed subscheme of the special fiber w... | https://mathoverflow.net/users/33009 | scheme of generalizations | Suppose for simplicity that $\mathcal O$ is a henselian (e.g., complete) DVR (otherwise, a point of $X\_\eta$ may have several specializations in $X\_s$) with field of fractions $K$. Consider $X=\mathbb P^1\_S$ parametriezd by a rational function $t$, and let $Z$ be the single point $t=0$ in the closed fiber. The the s... | 1 | https://mathoverflow.net/users/3485 | 127344 | 70,995 |
https://mathoverflow.net/questions/127345 | 0 | Let $X$ be a projective scheme over $S=spec A$, where A is a complete integrally closed noetherian local ring. $Y \subset X$ is a relative effective Catier divisor.
Then there exists a global section of $X$ over $S$ such that $Y$=image of this section. Is this true?
If not then it is true in which case?
Thank for yo... | https://mathoverflow.net/users/33052 | Question about relative Cartier divisor. | I am not sure if this question is appropriate for MO, but here is an answer.
The relative dimension, over $S$, of the image of a section equals $0$. So, if $Y$ is the image of a section, then $X\to S$ is smooth of relative dimension $1$ at every point of $Y$. Moreover, $Y$ then has relative degree $1$ over $S$. Conve... | 1 | https://mathoverflow.net/users/13265 | 127350 | 70,998 |
https://mathoverflow.net/questions/118508 | 0 | I have the following double integral:
$\int\limits\_0^x {\int\limits\_0^y {{e^{ - {K\_1}(u + v)}}{I\_0}\left( {2{K\_1}\sqrt {uv} } \right)dudv} }$
where $K\_1$ is a constant. Do you have any ideas of getting a closed form for this integral? Thank you very much.
| https://mathoverflow.net/users/30537 | Closed form for double integral? | I found the solution from a reference paper, which is: A Double Integral Containing the Modified Bessel Function: Asymptotics and Computation.
<http://www.ams.org/journals/mcom/1986-47-176/S0025-5718-1986-0856712-X/S0025-5718-1986-0856712-X.pdf>
| 1 | https://mathoverflow.net/users/30537 | 127356 | 71,000 |
https://mathoverflow.net/questions/127354 | -1 | On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is absolutely continuous with respect to $P$ that is, that for any Borel set $E$ of $\mathcal{X}$ we have $P(E)=0$ implies $Q(E)=0... | https://mathoverflow.net/users/33054 | Absolute continuity of probabilities on Polish spaces and open sets. | Your condition means precisely that the **support** of the measure $Q$ (the minimal closed set of full measure) is contained in the support of the measure $P$. Obviously it has nothing to do with absolute continuity.
| 1 | https://mathoverflow.net/users/8588 | 127360 | 71,002 |
https://mathoverflow.net/questions/127362 | 12 | I am looking for a counter example to the fact that a faithfully flat morphism is
an effective descent morphism for the category of quasi-coherent sheaves
when one forgets the quasi-compact hypothesis. If possible, I'd like a counter-example
on the *descent of morphism aspects*. In other words, I am looking for
a morph... | https://mathoverflow.net/users/9317 | Counter-example to faithfully flat descent | I think the following should be a counter-example.
Let $X=\text{Spec}(\mathbb{C}[T])$ be the affine line and $X'=\bigsqcup\_{x\in\mathbb{C}}\text{Spec}(\mathbb{C}[T]\_{(T-x)})$ be its faithfully flat cover built from all local rings at closed points. Let $F=\mathcal{O}\_{X}$ and $G=\bigoplus\_{x\in\mathbb{C}}\mathcal... | 12 | https://mathoverflow.net/users/2868 | 127373 | 71,007 |
https://mathoverflow.net/questions/127341 | 7 | As the title says.
Let $A^n$ be an $n$-dimensional closed Alexandrov space. Does it admit a bi-Lipschitz embedding into Euclidean space $\mathbb R^N$ for sufficiently large $N$?
I know there are some spaces that do not admit such an embedding; for example, a theorem by Pansu says that:
>
> The Heisenberg group
>... | https://mathoverflow.net/users/1190 | Does closed Alexandrov space admit a bi-Lipschitz embedding into $\mathbb R^N$? | See Distance embedding (27.5) in [our book](http://dl.dropboxusercontent.com/u/1577084/the-book.pdf)
| 8 | https://mathoverflow.net/users/1441 | 127374 | 71,008 |
https://mathoverflow.net/questions/127259 | 5 | We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, $\Omega$ is a bounded domain in $\mathbb{R}^n$ with $C^1$ boundary.
From [Wikipedia](http://en.wikipedia.org/wiki/Sobolev_inequ... | https://mathoverflow.net/users/31342 | Sharpness of the Sobolev embedding theorem | This response is closely related to my answer [Here](https://mathoverflow.net/questions/124028/what-goes-wrong-for-the-sobolev-embeddings-at-kn-p/124331#124331).
For the case $W^{2,n}$ we automatically get Holder regularity by applying Sobolev and then Morrey. However, we don't get Lipschitz. The following example "i... | 3 | https://mathoverflow.net/users/16659 | 127375 | 71,009 |
https://mathoverflow.net/questions/127348 | -1 | What is the (natural) bijection between the isomorphic class of sub modules and isomorphic class of quotient modules of a finitely generated torsion module over a PID.
Is there any inclusion relation between these classes?
| https://mathoverflow.net/users/33047 | Correspondence between submodules and quotient modules | If $M=R/(p^n)$ with $p$ prime, the result is clear. Since an arbitrary $M$ is a direct sum of such modules, the result is still clear.
| 0 | https://mathoverflow.net/users/10503 | 127387 | 71,014 |
https://mathoverflow.net/questions/127385 | 1 | Thompson's group may act by homeomorphisms on the circle.
Has this action a fixed point?
| https://mathoverflow.net/users/16804 | Orbits of Thompson's group | If you mean Thompson's $F$, please, specify the action on the circle.
If you mean Thompson's group $[F,F]$, the answer is yes, as it acts on the interval.
I think you are talking about Thompson's group $T$, and its dynamics on the circle was well described by Ghys and Sergiescu in
*[Sur un groupe remarquable de d... | 3 | https://mathoverflow.net/users/47274 | 127388 | 71,015 |
https://mathoverflow.net/questions/127390 | 1 | I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at the initial point $m\_0$ and this value can be any $H$-invariant antisymmetric poly-linear form on the tangent space $... | https://mathoverflow.net/users/nan | A question about G-Manifolds | This question is too elementary for this site. The idea is that we have to define $\omega$ at each point $m=gm\_0$ to be $\left(g^{-1}\right)^\* \omega\_0$. The $H$-invariance ensures that this result does not depend on the choice of a particular representative $g \in G$ that takes $m\_0$ to $m$. The fact that $G$ acts... | 5 | https://mathoverflow.net/users/13268 | 127392 | 71,016 |
https://mathoverflow.net/questions/127386 | 4 | What are the places where character table values of $S\_n$ occurs naturally? one such an example is when we write power sum symmetric function of order n in terms of Schur function of order n the coefficient of each monomial will be the character value of the respective conjugacy class and respective irreducible repres... | https://mathoverflow.net/users/33047 | Character table of Sn | My favourite place where the character table of $S\_n$ occurs naturally is in the enumeration of branched covers of Riemann surfaces (Hurwitz numbers).
In more detail, let $\Sigma$ be a closed Riemann surface of $g$ with $k$ marked points $x\_1, \ldots, x\_k$, and fix an integer $n\geq 1$. For each marked point $x\_i... | 3 | https://mathoverflow.net/users/7762 | 127393 | 71,017 |
https://mathoverflow.net/questions/127319 | 11 | A Riemannian metric on a manifold $X$ defines a function on the symplectic space $T^\*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?
| https://mathoverflow.net/users/7108 | Intuition for Levi-Civita connection via Hamiltonian flows | The intuition is that the Levi-Civita connection corresponds to the linearization of the geodesic flow plus a simple projective-geometric construction:
Let $c(t)$ be an orbit of the geodesic flow (projecting down to a geodesic), consider the vertical subspaces $V(t)$ along $c(t)$ and bring them back to the tangent sp... | 12 | https://mathoverflow.net/users/21123 | 127394 | 71,018 |
https://mathoverflow.net/questions/127403 | 1 | If $A$ is an abelian variety over a field, the Kummer variety $K\_A$ associated to $A$ is obtained as the quotient of $A$ by the involution $\iota: a \mapsto -a$. If $A$ is a surface, it is well-known that a resolution of $K\_A$ can be constructed simply by blowing up the image (under the quotient map) of the set of $2... | https://mathoverflow.net/users/33062 | Resolutions of higher-dimensional Kummer varieties | Yes, it is smooth. What is different from dim 2, is that the resolution is not crepant, so the resolution of $K\_A$ is not a Calabi--Yau variety.
| 2 | https://mathoverflow.net/users/4428 | 127411 | 71,022 |
https://mathoverflow.net/questions/123958 | 10 | My question is related to the following question by Mark Grant here on math overflow:
[Formal group law of unoriented cobordism](https://mathoverflow.net/questions/74770/formal-group-law-of-unoriented-cobordism)
There it is stated that $MO\_\*$ has a formal group law $F\_0$, universal for formal group laws in chara... | https://mathoverflow.net/users/28048 | A formal group law over oriented bordism | * MSO is certainly complex-oriented
* The resulting formal group law satisfies $[-1]\_F(x)=-x$
* Let $R\_\*$ be the universal example of a graded ring with a formal group with the above property, so there is a natural map $R\_\*\to MSO\_\*$. I think this is injective, and it becomes an isomorphism after inverting $2$.
... | 8 | https://mathoverflow.net/users/10366 | 127412 | 71,023 |
https://mathoverflow.net/questions/127371 | 8 | Restricting a quasi-conformal homeomorphism of the disc to the boundary gives a surjective homomorphism from $QC(D^2)$ (quasi-conformal homeos of $D^2$) to $QS(S^1)$ (quasi-symmetric homeos of the circle). Surjetivity here follows, for example, from the existence of natural extensions of QS homeos like Douady-Earle.
... | https://mathoverflow.net/users/5584 | quasi conformal, area preserving homomorphisms of the disc | Claim. A QS homeomorphism $f$ of the circle extends to a QC area preserving map of the disk if and only if $f$ is BL (bi-Lipschitz).
Proof.
1. Suppose that $f: S^1\to S^1$ is a QS map which admits an area-preserving quasiconformal extension $F: D^2\to D^2$. Then it immediately follows from the definition of quasi... | 7 | https://mathoverflow.net/users/21684 | 127414 | 71,024 |
https://mathoverflow.net/questions/127410 | 6 | Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $$C\_{\mathfrak{g}}(\xi)=\{\eta\in\mathfrak{g}:[\xi,\eta]=0\}, \quad C\_G(\xi)=\{g\in G:\mathrm{Ad}\_\mathfrak{g}(\xi)=\... | https://mathoverflow.net/users/25358 | Centralizers of nilpotent elements in semisimple Lie algebras | This determination of component groups goes back to Elashvili and Alexeevskii, but has been improved somewhat in a 1998 IMRN paper by Eric Sommers and a later joint paper by him and George McNinch [here](https://arxiv.org/abs/math/0204275). Your set-up is essentially equivalent to studying the same problem for a semisi... | 6 | https://mathoverflow.net/users/4231 | 127424 | 71,028 |
https://mathoverflow.net/questions/127423 | 18 | One has an $n$-dimensional convex polytope $P$ represented by an intersection of half-spaces:
\begin{equation}H\_i = \{ (x\_1,x\_2, \ldots,x\_n) \in \mathbb{R}^n \mid \sum\_{j=1}^n a\_{ij} x\_j \ge a\_{i0}, \ a\_{ij} \in \mathbb{R} \}, \ i = \overline{1,k} \end{equation}
\begin{equation}P = \{ X \in \mathbb{R}^n \mid... | https://mathoverflow.net/users/33027 | How many vertices can a convex polytope have? | Using $k$ half-spaces, the polytope has at most $k$ facets. For a fixed number of facets, the number of vertices is maximized, for example, by the [dual polytope](http://en.wikipedia.org/wiki/Dual_polytope) of the [cyclic polytope](http://en.wikipedia.org/wiki/Cyclic_polytope) with $k$ vertices. More generally, by the ... | 21 | https://mathoverflow.net/users/24076 | 127426 | 71,029 |
https://mathoverflow.net/questions/127419 | 3 | Let me explain what I mean:
* The width of the average person varies, perhaps with a normal distribution.
* Given a specific variance, how many people (on average) can sit side-by-side on a bench of a given length?
What would the graph of this look like? If Length is the x-axis, and Average Number of People is the ... | https://mathoverflow.net/users/33065 | The average number of people that can sit on a bench of a given length. | Assume that the process stops when someone can't fit.
I believe the distribution of the amount of overshoot is known as a *ladder height distribution*, and that this is in Feller's classic text, but I don't have that handy.
Let $f(x)$ be the expected number of people up to and including the first who is turned aw... | 7 | https://mathoverflow.net/users/2954 | 127430 | 71,031 |
https://mathoverflow.net/questions/127372 | 13 | Suppose we have some pointed connected topological space $X$. How can we determine if there exists a space $BX$, called *delooping* of $X$, such that its space of based loops $\Omega BX$ is homotopy equivalent to $X$? More generally, when can we deloop a space $n$ times?
Abstract nonsense
-----------------
There is... | https://mathoverflow.net/users/10605 | Proving that a space cannot be delooped. | The general question is indeed hard, but in view of Anton's side question to Dylan about compact Lie groups, I feel compelled to advertise a beautiful old result of John Hubbuck that everyone should know: If $X$ is a connected finite CW complex and a homotopy commutative $H$-space, then $X$ is homotopy equivalent (as a... | 17 | https://mathoverflow.net/users/14447 | 127431 | 71,032 |
https://mathoverflow.net/questions/127435 | 1 | What is the weak\* closure of {f:||f||=1}? I am sure this set is not closed in weak\* topology.
So what is the weak\* closure of this set. Thanks.
| https://mathoverflow.net/users/33047 | weak*closure of {f:||f||=1} in dual. | The weak$^{\*}$ closure of $\{f\in X^{\*}:\|f\|=1\}$ is the unit ball $\{f\in X^{\*}:\|f\|\leq 1\}$ for infinite dimensional Banach spaces $X$. To prove this, let $f\in X^{\*}$ be a functional with $\|f\|\leq 1$. Let $D$ be the set of all finite dimensional subspaces of $X$. Then $D$ is a directed set under inclusion. ... | 5 | https://mathoverflow.net/users/22277 | 127437 | 71,035 |
https://mathoverflow.net/questions/127434 | 7 | I have a map, constructed geometrically, $S^4 \to S^3$. I suspect that it is a representative for the generator $\eta\_3\in \pi\_4(S^3) \simeq \mathbb{Z}\_2$, but I am not 100% sure ($\eta\_3$ is defined as the suspension of the Hopf map $S^3 \to S^2$). I'd like to be able to detect the fact that my map is homotopicall... | https://mathoverflow.net/users/4177 | Detecting homotopy nontriviality of an element in a torsion homotopy group | How about thinking about framed cobordism, which in this case gives an isomorphism between $\pi\_4(S^3)$ and the group of cobordism classes of normally framed 1-manifolds in $S^4$. Since your map is constructed geometrically, it probably has a regular value. Pull this back to a collection of disjoint circles in $S^4$, ... | 22 | https://mathoverflow.net/users/23571 | 127439 | 71,036 |
https://mathoverflow.net/questions/127441 | 0 | Imagine I perform a random sequential adsorption (RSA) simulation for circles or discs of some radius $r \leq 1$ in $[0, 1]^2$ (I am open to changing this geometry to the unit circle). As a function of $r$, what is the best known lowerbound for the number of discs I can place on the surface s.t. all discs are guarantee... | https://mathoverflow.net/users/32871 | Random Sequential Adsorption of Discs on a Plane - What is the best known lowerbound for the number of circles (of some radius $r$) guaranteed to fit on $[0, 1]^2$? | If you can't fit another disc with radius $r$ in, then it means that every point is within $2r$ of one of the current centers. So, you are asking how efficient a covering by discs can be. Asymptotically, the [most efficient covering](http://mathworld.wolfram.com/DiskCoveringProblem.html) is the hexagonal one, with effi... | 1 | https://mathoverflow.net/users/2954 | 127442 | 71,037 |
https://mathoverflow.net/questions/127428 | 1 | Hi,
A part of [this](https://mathoverflow.net/questions/109989/concentration-results-for-inner-products-of-two-independent-random-gaussian-vecto/110099#110099) answer says that the inner product $\left( \frac{X}{|X|} \right)^T Y$, where X and Y are vectors with i.i.d. zero-mean Gaussian elements, is independent on X ... | https://mathoverflow.net/users/33067 | Inner product with normalized Gaussian | $X/|X|$ is almost surely a uniformly random element on the unit sphere of dimension $n-1$; this is not the same thing as a rotation, which is a matrix. Since a multivariate standard Gaussian vector is invariant in law under a fixed rotation, its law is certainly invariant under a random rotation as well: thus if $A$ is... | 5 | https://mathoverflow.net/users/4923 | 127443 | 71,038 |
https://mathoverflow.net/questions/33045 | 4 | I'm trying to estimate the product
$$\prod\_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea is that there are 4! congruence classes mod $(pqrs)^2$ such that $n,n+1,n+2,n+3$ are each congruent to 0 mod the square... | https://mathoverflow.net/users/6043 | Product over the primes | Modulo errors got unconditionally slightly smaller upper bound than your conjecture.
`prodeulerrat` from [Cohen's pari script](http://pari.math.u-bordeaux.fr/Scripts/) can compute $\prod\_{\text{p prime}}{1-1/O(x^2)}$.
For fixed $p < q < r < L$ define $g(p,q,r)=\prod\_{\text{s prime}}{1-24/(pqrs)^2}$
$g$ is efficie... | 3 | https://mathoverflow.net/users/12481 | 127451 | 71,040 |
https://mathoverflow.net/questions/127418 | 1 | I believe there is a reasonable notion of $\text{Ext}^1(G,H)$ in the category of groups (where $G$ and $H$ are groups). Is there a decent reference describing this?
My particular situation involves a nilpotent Lie group $G$ with finitely many components. One may form the short-exact sequence $$1\rightarrow G\_0\right... | https://mathoverflow.net/users/25358 | Extensions of Groups | Since $G$ is a Lie group with finitely many components, it has a maximal compact subgroup $K$, unique up to conjugation and $G=KG\_0$ (Mostow). Since $G$ is actually nilpotent, $K$ is unique and actually consists of the elements in $G$ contained in a compact subgroup. In particular $K$ is normal in $G$.
From $G$ we ... | 1 | https://mathoverflow.net/users/14094 | 127455 | 71,044 |
https://mathoverflow.net/questions/127447 | 2 | dear all, Let $H=Z\_2 \wr Z\_2 \wr...\wr Z\_2$ ( r times), I need to know the structure of H as a matrix group.
Thanks in advance
| https://mathoverflow.net/users/27932 | wreath product and matrix presentation | Since it came up in comments, I will give an answer about the group I believe was intended to be asked about. A monomial matrix is a matrix which has one non-zero entry in each row and one non-zero entry in each column. The monomial $n \times n$ matrices who non-zero entries are all $\pm 1$ form a group, which may be t... | 6 | https://mathoverflow.net/users/14450 | 127459 | 71,046 |
https://mathoverflow.net/questions/127475 | 5 | Let $\mathfrak{g}\supset \mathfrak{b}\supset \mathfrak{h}$ be a complex semisimple Lie algebra, with choice of Borel and Cartan subalgebras, $W$ the Weyl group.
W. Soergel's 'Endomorphismensatz' allows for the identification of $End\_{\mathcal{O}\_{0}}(P(w\_{0}))$ with the algebra of coinvariants $\mathbb{C}[\mathfr... | https://mathoverflow.net/users/9970 | Endomorphisms in Category O and Schubert Classes | 1) I believe there is such a description, though I think its pretty debatable whether it is likely to tell you very much about Schubert calculus.
Category $\mathcal{O}$ has a nice collection of objects called "tilting modules"; these are distinguished by have a Verma and dual Verma filtration (actually all of these a... | 3 | https://mathoverflow.net/users/66 | 127486 | 71,055 |
https://mathoverflow.net/questions/127484 | 3 | It would seem as though the sentence s(E) which expresses the existence of a non-trivial
elementary embedding of the universe V into itself-and which can be formalized in the
first order language of NBG-could also be formalized in the language of Quine's NF. These
two set theories would seem to share the same first or... | https://mathoverflow.net/users/4423 | A question about Kunen's inconsistency theorem | This is more of a comment than an answer, but it is too long to
fit in a comment box.
There are a number of subtle issues concerning your claim that one
may formalize the Kunen inconsistency as an assertion in the
first-order language of set theory. Kunen himself formalized his
theorem as a second-order assertion in ... | 2 | https://mathoverflow.net/users/1946 | 127487 | 71,056 |
https://mathoverflow.net/questions/127409 | 2 | I recently learned of a result of Brundan describing the centre of the regular block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}\_{n}$ as the cohomology of a corresponding Springer fibre (available [here](http://darkwing.uoregon.edu/~brundan/papers/springernew.pdf)); this is an isomorphism at the level of $\... | https://mathoverflow.net/users/9970 | Springer Action on Centre of Parabolic Category O (after Brundan) | David's comment is correct. This is proven in Section 3 of this paper of Stroppel: <http://arxiv.org/abs/math.RT/0608234>
| 1 | https://mathoverflow.net/users/66 | 127488 | 71,057 |
https://mathoverflow.net/questions/127473 | 15 | I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, a conformal compactification of a manifold $M$ is a compactification $\tilde{M}$ in which all conformal transformations... | https://mathoverflow.net/users/22337 | How unique is a conformal compactification? | For Lorentzian manifolds, the conformal completion need not be compact. A typical example is the universal covering of the $d$-dimensional anti-de Sitter space-time (the maximally symmetric solution of the vacuum Einstein equations with negative cosmological constant) - its conformal boundary is diffeomorphic to $\math... | 12 | https://mathoverflow.net/users/11211 | 127492 | 71,059 |
https://mathoverflow.net/questions/127377 | 3 | Let $A$ be a ${\bf valuation}$ ${\bf ring}$ in the classical sense: $A$ is a domain with quotient field $K$ and for every non-zero $x\in K$ one has $x\in A$ or $x^{-1}\in A$.
Now ${\bf Bourbaki}$ (Commutative Algebra, Chapter VI, exercise 1 for §4) suggests that if $\mathfrak{p}$ is any non-maximal prime ideal of $A... | https://mathoverflow.net/users/31923 | Primary ideals in valuation rings | There are certainly valuation rings in which each non-maximal prime ideal has no primary ideal except itself. Any Noetherian valuation ring has this property for trivial reasons. Also, this is true for any valuation ring $R$ with value group of the form $G\_1 \oplus \cdots \oplus G\_n$ under lexicographic order, where ... | 0 | https://mathoverflow.net/users/19045 | 127494 | 71,060 |
https://mathoverflow.net/questions/127028 | 11 | I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A\_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in the case underlying chain complexes are unbounded, ungraded.
Related question, let $M$ denote the category of small $... | https://mathoverflow.net/users/20196 | Model structure on the category of small $A_\infty$ categories, hocolims. | Look at Lefevre-Hasegawa PhD Thesis (unfortunately in French) at <http://arxiv.org/abs/math/0310337>, where this is done. For a reference in English, you can look there: <http://math.unice.fr/~brunov/HomotopyTheory.pdf> , where you have to restrict to the the Koszul operad $As$ of associative algebras and then consider... | 9 | https://mathoverflow.net/users/12352 | 127502 | 71,062 |
https://mathoverflow.net/questions/127425 | 3 | I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the entries of $M$ are integers, and the entries of the $i$th row of $M$ are bounded by some positive parameter $k\_i$. Then by expressing the de... | https://mathoverflow.net/users/10898 | Average size of determinants of integer matrices? | As noted in Will's comment above, it's easy to compute the expected square of the determinant. More precisely, we have
$$E(\det M^2)=n! \prod \frac{k\_i (k\_i+1)}{3}.$$
Let $M'$ be formed from $M$ by dividing each row by $\left(\frac{k\_i(k\_i+1)}{3}\right)^{1/2}$. Now each entry has mean $0$ and variance $1$, and f... | 3 | https://mathoverflow.net/users/405 | 127513 | 71,066 |
https://mathoverflow.net/questions/127520 | 56 | The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved. It has by now very many analogues, many of them still open. Are there important analogues that are now known to be false?
| https://mathoverflow.net/users/38783 | Are there refuted analogues of the Riemann hypothesis? | There is a well-known example of Davenport and Heilbronn of a Dirichlet series that in some sense is not *so* different from the Riemann-zeta function but that has zeros off the critical line.
The function is defined
$$\sum\_{n=1}^{\infty} \frac{a\_n}{n^s}$$
where $a\_n$ equals $1, c, -c, -1, 0$ for $n$ equal to $1... | 55 | https://mathoverflow.net/users/nan | 127522 | 71,070 |
https://mathoverflow.net/questions/127532 | 1 | We know about the [Carathéodory's theorem](http://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29) which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex objects of any real manifold?
I beli... | https://mathoverflow.net/users/651 | Extensions of Carathéodory's theorem | The statement of the theorem only uses the linear structure of the ambient space, so the Banach space structure does not affect its validity.
One way of proving the theorem is by applying Helly's theorem. The latter seems to be more readily amenable to generalisations because of obvious connections to topology.
| 1 | https://mathoverflow.net/users/28128 | 127537 | 71,075 |
https://mathoverflow.net/questions/127529 | 2 | Though I am in a situation considering only local-zeta integral, to explain my question briefly, let me ask it in quite general form.
Let $f(s,g)$ be a two variable smooth good (in a suitable sense) function and let $F(s)=\int\_{G}f(s,g)dg$. Assume $F(s)$ is absolutely convergent for $\Re(s)>0$ and has meromorphic co... | https://mathoverflow.net/users/29334 | On the absolute convergence of the local-zeta integral. | No to the stronger statement. Here is the first simple example. If you take the zeta integral of a function f and a non trivial Dirichlet character at s, then take absolute values inside the integral you obtain the zeta integral of |f| with s`=Re s. The former has no pole at zero but the later. So the order of divergen... | 0 | https://mathoverflow.net/users/10400 | 127544 | 71,076 |
https://mathoverflow.net/questions/127540 | 8 | I originally posted this question to [MSE](https://math.stackexchange.com/questions/325621/does-induction-for-a-functor-algebra-imply-it-is-initial) but there were no answers except for a partial one from me, so I'm trying again here. I'm an undergrad, not a researcher, so forgive me (and correct me!) if my terminology... | https://mathoverflow.net/users/26698 | Does "induction" for a functor algebra imply it is initial? | It looks as though you might have already observed this yourself, but suppose $F: C \to C$ is an endofunctor and $C$ has equalizers. Then if $F$ has a *weakly* initial algebra $X$ (meaning that for every $F$-algebra $Y$ there exists an $F$-algebra map $X \to Y$), then $X$ is initial if and only if every $F$-subalgebra ... | 6 | https://mathoverflow.net/users/2926 | 127546 | 71,078 |
https://mathoverflow.net/questions/127499 | 5 | Hello,
I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L\_1$ distance is one.
Now I want to claim something like this:
For any partition of $V$ into $V\_1, V\_2$ with $n/4\leq |V\_1| \leq n/2$, there are at least $\sqrt{n}$... | https://mathoverflow.net/users/33084 | Perimeter/Neighborhood of a graph on grid | If you are serious about this, search the web for the "edge-isoperimetric problem
for the grid graph". If you just want a (relatively) short solution to your specific problem, consider the following.
Let $k:=\sqrt n$, and assume for simplicity that $k$ is an integer and
$|V\_1|=n/4=k^2/4$. Let $x\_1,\ldots,x\_k$ and ... | 2 | https://mathoverflow.net/users/9924 | 127555 | 71,083 |
https://mathoverflow.net/questions/127567 | 2 | All varieties are assumed over $\mathbb{C}$. Consider a geometrically ruled surface $X$ over a curve $C$, it is known that $X$ can be realized as the projective bundle associated to a rank $2$ vector bundle $E$ over $C$, that is $X=\mathbb{P}(E)$.
But there are two ways to associate a projective space $\mathbb{P}(V)... | https://mathoverflow.net/users/15289 | Are these two definitions of $\mathcal{O}(1)$ over a ruled surface closely related? | If $E$ is a rank 2 bundle then $E^\* \cong E \otimes \Lambda^2E^\*$. This gives the required relation for $O(1)$ bundles.
| 4 | https://mathoverflow.net/users/4428 | 127575 | 71,091 |
https://mathoverflow.net/questions/127576 | 4 | What exactly do mathematicians mean when they refer to "the data" involved in a construction?
I've encountered this many times and I can usually figure out what's going on, but I am curious about the terminology, how it was introduced, and its connotations.
A concrete example: In pages 35-6 of "Differential Geometr... | https://mathoverflow.net/users/29961 | What is "Data" involved in a mathematical construction? | I don't know. However, this time I won't let that stop me from answering.
If you talk to a carpenter or a craftsman about a geometric or algebraic
construction, they might look at you in a funny way, since your product
is not material. Also (unless you are doing topological surgery or concatenation
of words), you usu... | 4 | https://mathoverflow.net/users/3528 | 127580 | 71,093 |
https://mathoverflow.net/questions/127571 | 1 | I am working through Lieb/Loss's "Analysis", and have been stuck on one of the problems for a while;
Suppose we are on $\mathbb{R}^n$ and define $f(x) = |x|^{-n}$. This is not a locally integrable function. However if $\phi \in C\_c^{\infty}(\mathbb{R}^n)$ is a function vanishing at the origin, we can still define th... | https://mathoverflow.net/users/33110 | Nonintegrable inverse powers as distributions | The related topic here is the homogeneous distribution on $\mathbb{R}^n\0$ and its extension to $\mathbb{R}^n$. In your case $T\_{f}$ is a homogeneous distribution on $\mathbb{R}^n\0$ of degreee $-n$. And it's always possible to extend it to a distribution on $\mathbb{R}^n$,which may not be homogeneous any more. In fac... | 1 | https://mathoverflow.net/users/23078 | 127583 | 71,094 |
https://mathoverflow.net/questions/127531 | 2 | Let $\mathcal{G}=(A\rightrightarrows X)$ be a groupoid.
Here $X={\rm Ob}(\mathcal{G})$, $A={\rm Ar}(\mathcal{G})$,
and we have 5 maps:
$s,t\colon A\to X$ (the source and the target, surjective),
$m\colon A\times\_X A\to A$ (multiplication of composable arrows),
${\rm id}\colon X\to A$ ($x\mapsto{\rm id}\_x$, injective)... | https://mathoverflow.net/users/4149 | Constructing a stack (gerbe) from a connected groupoid | The thing that Simon gives in his first comment is just a prestack, it needs to be stackified. You can do this by letting $\mathbb{G}(S)$ be the groupoid of principal $\mathcal{G}$-bundles in $\Gamma Set$, in other words, $\Gamma$-equivariant $\mathcal{G}$-bundles in Set.
In more detail, I'm assuming you are consider... | 3 | https://mathoverflow.net/users/4177 | 127595 | 71,100 |
https://mathoverflow.net/questions/127601 | 18 | Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see <http://adsabs.harvard.edu/abs/1947pqm..book.....D> for the exact reference (but no text). What is the best way of formalizing this to a mathematician's satisfaction?
| https://mathoverflow.net/users/28128 | Does the derivative of log have a Dirac delta term? | In integral form, this amounts to the [Sokhotski-Plemelj](http://en.wikipedia.org/wiki/Sokhotski-Plemelj_theorem) theorem:
$\lim\_{\epsilon\rightarrow 0^{+}}\int\_{-\infty}^{\infty}dx f(x) \frac{d}{dx}\log (x+i\epsilon)=-i\pi f(0)+{\cal P}\int\_{-\infty}^{\infty}dx f(x)\frac{1}{x}$.
The symbol ${\cal P}$ indicates... | 19 | https://mathoverflow.net/users/11260 | 127604 | 71,103 |
https://mathoverflow.net/questions/127204 | 6 | Consider a continuous Markov chain $X = (X\_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from each state, there can be transitions to 2 other states only).
Let $P\_t$ be the transition probability matrix, so $P\... | https://mathoverflow.net/users/33007 | Efficient computation of Markov chain transition probability matrix | What you need is called "computing the action of the matrix exponential" (that is, computing $\exp(A)b$ without forming $\exp(A)$ explicitly. There are techniques based on complex integrals and Krylov subspaces. See <http://eprints.ma.man.ac.uk/1426/> and the references included there.
| 1 | https://mathoverflow.net/users/1898 | 127626 | 71,112 |
https://mathoverflow.net/questions/127462 | -4 | Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily infinitely many real zeros. Let us consider the functions: $f^{(k)}(1-2\prod\_{j=1}^{k}t\_{j})$ for $k=1,..,r$ and $(t\_... | https://mathoverflow.net/users/25947 | How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$? | As your question is stated, nothing guarantees that $f^{(k)}$ has a single simple zero. The fact that you introduce extra paremeters cannot change that fact! So, in general, the answer is: there is no way.
| 3 | https://mathoverflow.net/users/24309 | 127632 | 71,115 |
https://mathoverflow.net/questions/127607 | 1 | Where can I find a proof for the Schur-Cohn stability test?
| https://mathoverflow.net/users/31955 | Schur-Cohn Stability Test. | A reference is: Proakis J.G., Manolakis D.G., “Digital Signal Processing”, Prentice Hall, 1996.
| 2 | https://mathoverflow.net/users/23542 | 127638 | 71,119 |
https://mathoverflow.net/questions/127633 | 41 | Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which exists in a world where standard conjectures and anything else you can expect is true, in the spirit of Grothendieck, and ... | https://mathoverflow.net/users/9317 | What are the possible motivic Galois groups over $\mathbb Q$? | The first question (applied to $\mathrm{GL}(2)$-abelian varieties over $\mathbf{Q}$) seems to include the following problem: what totally real fields $F$ occur as the field of coefficients of a classical weight $2$ modular form? This seems a totally impossible question to answer.
For example, it includes the question o... | 16 | https://mathoverflow.net/users/33127 | 127642 | 71,120 |
https://mathoverflow.net/questions/127624 | 2 | When $F = \mathbb{R}, \mathbb{C}$ or $\mathbb{H}$, there are fibrations $$O(k,F)\rightarrow V\_k(F^n)\rightarrow G\_k(F^n)$$ where $V\_k(F^n)$ are Steifel manifolds and $G\_k(F^n)$ are Grassmannians. When $k=1$ these reduce to the Hopf fibrations $$S^{d-1}\rightarrow S^{dn-1}\rightarrow FP^{n-1}$$ where $d=\text{dim}F$... | https://mathoverflow.net/users/9563 | Geometry of Hopf fibrations and the fibration of Steifel Manfiolds over Grassmannians | You are right, this is true. You start with the invariant metric on $O(k,F)$. Then the relevant subgroups act isometrically, thus you get Riemannian submersions.
The Grassmann manifolds are symmetric spaces, but the Stiefel manifolds are only homogeneous.
A very good source which also gives explicitly all geodesics an... | 2 | https://mathoverflow.net/users/26935 | 127644 | 71,121 |
https://mathoverflow.net/questions/127560 | 6 | Here I describe the sort of reference I'm after with a motivating example. I am **not** seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good references on solving equations of this sort.
I had an equation
>
> $2 \lfloor a \rfloor\_{c} - a = d - c$
> ... | https://mathoverflow.net/users/1536 | References on techniques for solving equations with discontinuous functions such as floor and ceiling? | Joe Roberts provided the words for the calligraphed book Elementary Number Theory: A Problem Oriented Approach, which was printed in the 1970's. This book has a chapter on brackets, which in some of the number theory literature is an older name for one or both of the floor and ceiling functions. While not providing as ... | 2 | https://mathoverflow.net/users/3528 | 127652 | 71,126 |
https://mathoverflow.net/questions/109444 | 26 | Put $X=\mathbb{H}P^\infty$ (so $X$ classifies quaternionic line bundles, and $\Omega X=S^3$). There is no obvious reason for $X$ to be an H-space, because the tensor product of quaternionic vector spaces is not naturally a quaternionic vector space. Below I will prove that there is no nonobvious H-space structure. Howe... | https://mathoverflow.net/users/10366 | Is $\mathbb{H}P^\infty_{(p)}$ an H-space? | No. If it were an $H$-space, there would be self maps of $\mathbb{H}P^\infty\_{(p)}$ inducing multiplication by $k$ in degree $4$ homology for all integers $k$. But this is not the case by a Theorem of S. Feder and S. Gitler in "Mappings of quaternionic projective spaces", Bol. Soc. Mat. Mex. 34 (1975) 12-18. Using Ada... | 19 | https://mathoverflow.net/users/33141 | 127659 | 71,128 |
https://mathoverflow.net/questions/127660 | 7 | We know that group cohomology $H^2(G,U(1))$ consists of 2-cocycles $\beta(A,B)\in U(1)$ corresponding to elements in the group $H^2(G,U(1))$, where $A\in G,B \in G$. Note that $\beta(A,B)$ satisfies 2-cocycles conditions: $\frac{\beta(A,B)\beta(AB,C)}{\beta(A,BC)\beta(B,C)}=1$, with $A,B,C\in G$.
For example,
(1)$... | https://mathoverflow.net/users/27004 | Explicit 2-Cocycles of G=Z2×Z2xZ2 over U(1) | It turns out that by playing around the $U(1)$ form of 2-cocylces, I manage to provide some answers to (2) and (3) and partially (4).
For (2)$H^2(Z\_2^2,U(1))=Z\_2$,
the 2-cocycles are $\beta(b,c)=\beta\_1^{n\_1}=\exp({i\pi}n\_1(b\_1 c\_2))$, with $b=(b\_1,b\_2)\in Z\_2^2$, $c=(c\_1,c\_2)\in Z\_2^2$. Here $b\_1,b\_2... | 7 | https://mathoverflow.net/users/27004 | 127671 | 71,135 |
https://mathoverflow.net/questions/127657 | 13 | In Physics one often encounters maps from a certain manifold $M$ to a Lie group $G$. For example, in gauge theories, this gives a gauge transformation, wich is a symmetry of a theory. It is then important to give a homotopy classification of such maps. One of the physical motivations is that maps non-homotopic to const... | https://mathoverflow.net/users/32985 | Homotopy classes of maps to Lie groups | If the dimension of $M$ is low relative to that of $G$ then the calculation of $[M,G]$ typically reduces to stable homotopy theory or generalised cohomology, for which many methods are known. For example, if $\dim(M)<2n$ then
$$[M,U(n)]\simeq [M,U(\infty)] \simeq [\Sigma M,BU(\infty)] \simeq K(\Sigma M) $$
(where $K$ ... | 15 | https://mathoverflow.net/users/10366 | 127683 | 71,138 |
https://mathoverflow.net/questions/127615 | 5 | An old problem: to show that *every bounded left integrable function is also Riemann integrable*.
I know some (for me) not elementary proofs: [Gillespie](http://www.jstor.org/stable/2007121) and [Kristensen et al](http://www.jstor.org/stable/2311188) (theorem 1).
The question: can it be an undergraduate problem, ... | https://mathoverflow.net/users/33124 | Cauchy's left endpoint integral (1823) | The following argument may not be the most direct one, but follows as a quick consequence of the characterization of Riemann integrability, which is perhaps the main result of the theory.
For a bounded function $f:[a,b]\to\mathbb{R}$ and for $\lambda >0$ let's denote
$$J \_ \lambda :=\{ x\in [a,b]\, : \,\limsup \_{y... | 6 | https://mathoverflow.net/users/6101 | 127684 | 71,139 |
https://mathoverflow.net/questions/127687 | 4 | Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot , \cdot)$ the Kähler form. How can one show:
1. If $\pi\_{1}(M)=0$ and $Ric(g)=0$ then the canonical bundle $K\_{M} :... | https://mathoverflow.net/users/32980 | Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold | The answer to the first question is that the Ricci tensor defines a (1,1) form (called the Ricci form) and this is the curvature of the connection on $K\_M$ induced by the Levi-Civita connection on $T M$. So if the manifold is Ricci-flat, $K\_M$ is flat and hence if $M$ is simply-connected it is trivial. Flatness says ... | 6 | https://mathoverflow.net/users/394 | 127689 | 71,141 |
https://mathoverflow.net/questions/127670 | 3 | I have a torsion-free non-abelian nilpotent group $\Gamma$ of cohomological dimension $n$. Is it possible to say anything about the number of generators of $\Gamma$ in a minimal presentation?
Can I assume that the number of generators can be chosen to be less than $n$?
| https://mathoverflow.net/users/11084 | Cohomological dimension of groups & number of generators | Finitely generated torsion-free nilpotent groups are polycyclic. Therefore, their cohomological dimension equals their Hirsch length.This is a result of Gruenberg. One can find it in Gruenberg's book 'Cohomological topics in group theory' in section 8.8 or in Robert Bieri's Book on 'homological dimension of discrete gr... | 9 | https://mathoverflow.net/users/31670 | 127695 | 71,144 |
https://mathoverflow.net/questions/127679 | 3 | I have come across the following reference to D K Faddeev's construction of quaternionic fields in the book *The Embedding Problem in Galois Theory* by Ishkhanov, Lur'e and Faddeev :
[45] D. K. Faddeev, *Construction of algebraic domains whose Galois group is the quaternionic group*, Leningrad. Gos. Univ. Uchen. Zap.... | https://mathoverflow.net/users/2821 | D K Faddeev's construction of quaternionic fields | I don't have the paper you refer to, but on the page <http://www.math.spbu.ru/vestnik/2008/vestnik0801/dfaddeev.pdf>, which is dedicated to the 100th birthday of Faddeev, his early work is described in the following way (3rd paragraph of the second page):
"The second direction which interested Faddeev in the first ye... | 5 | https://mathoverflow.net/users/3272 | 127701 | 71,146 |
https://mathoverflow.net/questions/127456 | 11 | In 1965 Shepherdson proved that FLT is independent of the fragment of PA that uses only open induction and signature $0,S,+\times$. Indeed $2x+1\neq 2y$ is independent of that fragment. Schmerl gives a good general criterion for independence from that fragment in ``Diophantine equations in a fragment of number theory''... | https://mathoverflow.net/users/38783 | Does any lower bound on proofs of FLT improve Shepherdson 1965? | [Leszek Kołodziejczyk](http://dx.doi.org/10.1016/j.apal.2011.06.003) has devised a method how to extend some type of Shepherdson-like models of IOpen into models of Buss’s theory $T^0\_2$ (a weak subsystem of $I\Delta\_0+\Omega\_1$). In particular, he has shown that $T^0\_2$ does not prove that $x^3+y^3=z^3$ has no non... | 4 | https://mathoverflow.net/users/12705 | 127705 | 71,149 |
https://mathoverflow.net/questions/127706 | 2 | I hope somebody can give me a good reference for the following:
Let $G$ be a complex reductive group $H$ be a closed subgroup. Let further $R$ be any $\mathbb{C}$-algebra. Then the canonical map
$$G(R)/H(R)\to (G/H)(R)$$
is known to be injective but in general not surjective. See for example [1].
### So now my que... | https://mathoverflow.net/users/32972 | A question about $R$-points of an complex reductive group. | By Hilbert's Theorem 90, every torsor for a split torus over a field is trivial. Thus, as pranavk has commented, this should give surjectivity.
$\textbf{Edit.}$ The first answer I wrote (now changed) applied to the full center $Z$. I did not realize that the OP is asking about the quotient by $Z\_e$, the connected co... | 2 | https://mathoverflow.net/users/13265 | 127707 | 71,150 |
https://mathoverflow.net/questions/127711 | 4 | Is it true that if a finite CW complex $X$ is simply connected, and $\tilde{H}\_i(X, \mathbb{Q}) =0$ for $i \neq D$, then $X$ is rationally homotopy equivalent to a bouquet of $D$-dimensional spheres?
(In my setting $D \ge 3$, in case that makes any difference.)
| https://mathoverflow.net/users/4558 | sufficient conditions for rational homotopy equivalence | Yes. By a generalization of the Hurewicz theorem (which can be formulated more generally in terms of Serre classes), if $X$ is simply connected and has trivial rational homology below dimension $D$, then $\pi\_i(X)\otimes\mathbb{Q}=\tilde{H}\_i(X,\mathbb{Q})$ via the Hurewicz map for all $i\leq D$. In particular, for $... | 7 | https://mathoverflow.net/users/75 | 127716 | 71,153 |
https://mathoverflow.net/questions/125861 | 25 | Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a long period, but I
I can not find an accurate proof. So if somebody can help me,
I will appreciative this very much.
... | https://mathoverflow.net/users/20491 | $f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$ | The following papers prove this:
MR0682456 Reviewed Joris, Henri Une C∞-application non-immersive qui possède la propriété universelle des immersions. (French) [A nonimmersive C∞ mapping having the universal property of immersions] Arch. Math. (Basel) 39 (1982), no. 3, 269–277.
MR0833407 Reviewed Duncan, John; Kran... | 17 | https://mathoverflow.net/users/26935 | 127724 | 71,155 |
https://mathoverflow.net/questions/127719 | 5 | Where can I find a readable textbook or lecture notes on asymptotic expansions ?
| https://mathoverflow.net/users/33165 | textbooks on asymptotic expansions | De Bruijn's "Asymptotic methods in analysis" is an excellent book for beginners. You'll need to work through it diligently to learn everything but no advanced a priori knowledge is required. Also, you can easily download it from many online places that do not worry too much about copyright and, even if you decide to st... | 10 | https://mathoverflow.net/users/1131 | 127725 | 71,156 |
https://mathoverflow.net/questions/127726 | 4 | The precise question is this: given a irrational $r\in \mathbb{R}$, is it true that $r\mathbb{Z}$ is dense (topologically) in $\mathbb{R}/\mathbb{Z}$?
The reason this came up was actually a teaching moment for Calc II; I wanted to use $sin(n)$ as an example of a bounded divergent sequence, but I was actually not sure... | https://mathoverflow.net/users/15735 | Integer multiples of a irrational dense in R/Z ? | Yes. For elementary reasons.
Suppose it weren't dense. Then there would be some little interval not hit, of some positive length say $1/N$.
But this cannot happen. Divide the circle into N little equal intervals and consider $0,r,2r,...,Nr$. By pigeonhole some two of them lie in the same interval, say $ar,br$ (and ... | 9 | https://mathoverflow.net/users/8080 | 127728 | 71,157 |
https://mathoverflow.net/questions/127676 | 3 | The question comes from my attempt to understand the following question.
[height of contracted prime ideals in power series rings](https://mathoverflow.net/questions/126840/height-of-contracted-prime-ideals-in-power-series-rings)
$\bullet$ My original question: Let $(R,m)$ be a Noetherian local ring and $\hat{R}$ the... | https://mathoverflow.net/users/22388 | Dimension of formal fiber | For question 1, my geometric explanation is this: you can find an irreducible "analytic curve" $C$, i.e. a 1-dimensional closed subscheme of $X$, which is "as transcendental as possible", meaning that it is not contained in any algebraic hypersurface (i.e. does not map to a hypersurface in $Y$). Now if $p$ is the gener... | 5 | https://mathoverflow.net/users/7666 | 127738 | 71,160 |
https://mathoverflow.net/questions/127737 | 2 | Is there a smooth modular compactification of the moduli space of smooth curves of genus $ g > 1 $ over $ \mathbb{C} $?
I am willing to allow for enrichments such as level structures. The compactification should be a projective variety rather than a stack. Any references are highly appreciated.
| https://mathoverflow.net/users/16102 | smooth modular compactification of moduli of curves | Sure: see Eduard Looijenga, "Smooth Deligne-Mumford compactifications by means of Prym level structures", which completely answers your question.
There is also later work by de Jong-Pikaart, Boggi-Pikaart and Abramovich-Corti-Vistoli where more general non-abelian level structures are considered, and over more genera... | 3 | https://mathoverflow.net/users/1310 | 127742 | 71,163 |
https://mathoverflow.net/questions/127745 | 1 | For Alexandrov manifold in the title we mean 3-dim Alexandrov apace which is also a topological. manifold.
Shioya-Yamaguchi posted a conjecture on their paper "Collapsing 3-manifold with lower sectional curvature bound"
>
> Any three-dimensional compact, simply connected, nonnegatively curved Alexandrov space with... | https://mathoverflow.net/users/30176 | 3-dim 1-connected Alexandrov manifold with curvature $\ge 0$ Heomomorphic to sphere? | The Poincare conjecture in dimension 3 tells us that any 3-dimensional compact simply connected topological manifold is diffeomorphic to the 3-sphere. Surely you don't want to assume that your space is a manifold? Is its dimension as a manifold perhaps larger than 3?
| 1 | https://mathoverflow.net/users/13268 | 127750 | 71,167 |
https://mathoverflow.net/questions/123493 | 23 | Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct characterization of a Gaussian measure which does not rely on finite-dimensional projections? This definition is analogou... | https://mathoverflow.net/users/238 | What is a Gaussian measure? | You could alternatively try defining Gaussian measures as [$2$-stable](http://en.wikipedia.org/wiki/Stable_distribution) distributions. This does remove any reliance on finite dimensional projections, and even removes reference to topology. Let $V$ be a measurable vector space (by which, I mean a real vector space $V$ ... | 13 | https://mathoverflow.net/users/1004 | 127752 | 71,169 |
https://mathoverflow.net/questions/127646 | 1 | Hi,
let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter.
I want to show that $\gamma$ is strictly positive definite function, i. e., for $n\in\mathbb{N}$ the matrix $\Gamma = (\gamma(i-j))\_{i,j=1}^n$ is (st... | https://mathoverflow.net/users/33133 | Strictly positive definite autocovariance function of fGn | As per Mathematica, the inverse Fourier transform of your function $\gamma(x)$ is
\begin{equation\*}
\frac{\sqrt{2 \pi } \sec \left(\frac{\pi h}{2}\right) (\cos (t)-1) |t|^{-h-1}}{\Gamma (-h)}
\end{equation\*}
But for $0 < h < 1$, $\sec(\pi h/2) / \Gamma(-h)$ is negative, also $\cos(t)-1 \le 0$, so overall the IFT ... | 1 | https://mathoverflow.net/users/8430 | 127753 | 71,170 |
https://mathoverflow.net/questions/127754 | 1 | Consider a function f continuous on a compact interval.
Approximate it by a sequence of polygonal functions (you can).
Then consider a sequence of primitives of the polygonal functions (you can).
At last consider the limit of the latter sequence (you can).
Now you have found a primitive of f (you know) *without... | https://mathoverflow.net/users/33124 | Newton integration without integration | 1. What does this procedure have to do with Newton? Just curious.
2. If you approximate with "polygonal" (=piecewise linear) functions in the most natural way,
that is take points $(x\_k,f(x\_k))$ and connect them with straight line segments, what you obtain
is the "trapezoid rule" for approximate evaluation of integra... | 2 | https://mathoverflow.net/users/25510 | 127757 | 71,173 |
https://mathoverflow.net/questions/127730 | 11 | (From [MSE](https://math.stackexchange.com/questions/363522/what-is-the-correspondence-between-combinatorial-problems-and-the-location-of-th))
In the wikipedia [article](http://en.wikipedia.org/wiki/Gian-Carlo_Rota) on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one ... | https://mathoverflow.net/users/93724 | What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called? | Rota himself called this correspondence "the critical problem". You can find the full quote in Michael Lugo's [blog](http://godplaysdice.blogspot.nl/2007/11/asymptotics-of-partition-polynomials.html).
As explained by Garrett Birkhoff (in his book on Lattice Theory), the critical problem consists in locating the zeros... | 8 | https://mathoverflow.net/users/11260 | 127758 | 71,174 |
https://mathoverflow.net/questions/127712 | 2 | Let $f \colon U \to \mathbb{R}$ be a twice differentiable function, where $U$ is an open subset of $\mathbb{R}^n$. Here twice differentiable means that all the second partial derivatives $\frac{\partial}{\partial x\_i} (\frac{\partial}{\partial x\_j} f)$ exist (however they are not necessarily continuous). Suppose $\De... | https://mathoverflow.net/users/22804 | Nonharmonic solutions of Laplace's equation | Here's a counterexample in $n\ge3$ dimensions. With $\lVert\cdot\rVert$ denoting the Euclidean norm, set
$$
f(x)=\begin{cases}
x\_1x\_2x\_3\lVert x\rVert^{-n-4},&{\rm if\ }x\not=0,\cr
0,&{\rm if\ }x=0.
\end{cases}
$$
This is harmonic on $x\not=0$, has first and second order derivatives at the origin, but $f$ along with... | 3 | https://mathoverflow.net/users/1004 | 127764 | 71,178 |
https://mathoverflow.net/questions/127765 | 2 |
>
> Set D : Set of decision algorithms
> X∈D if and only if
>
>
> 1. X is an Turing machine algorithm with finite length
> 2. takes one input i, binary number
> 3. X(i)=0 or X(i)=1 or X(i) runs forever
>
>
>
Definition: equivalent
Algorithm X∈D and Y∈D are equivalent if and only if
For all i
X(i) =1 ⇔... | https://mathoverflow.net/users/33177 | existence of equivalence checking algorithm | There is no computable equivalence checker. The reason is that if there were, we could solve the halting problem, as follows: given a Turing machine program $p$ and input $x$, design another program $X$ that on any input $i$ first runs $p$ on $x$, and if this halts, then outputs $1$. So $X$ is equivalent to the always-... | 4 | https://mathoverflow.net/users/1946 | 127766 | 71,179 |
https://mathoverflow.net/questions/127717 | 20 | Let me detail the title of the question. I'm trying to give students an intuition of what the class number is.
Let $K=\mathbb{Q}(\sqrt{-d})$, with $d>0$ a square-free integer, be a quadratic imaginary field. Let $\mathcal{O}\_K$ be its ring of integers. It is of the form $\mathbb{Z}[\tau]$ with $\tau=\sqrt{-d}$ or $\... | https://mathoverflow.net/users/33163 | how to visualize the class number of an imaginary quadratic field? | This is an interesting question that I've wondered about myself, so I can't really answer it properly but I'll make a couple elementary observations. First, for a lattice in ${\mathbb Z}[\tau]\subset{\mathbb C}$ to be an ideal just means that multiplication by $\tau$ takes the lattice to itself. For example, for the Ga... | 23 | https://mathoverflow.net/users/23571 | 127784 | 71,185 |
https://mathoverflow.net/questions/127779 | 11 | Let $G$ be a compact Lie group, and $g$ its associated Lie algebra.
In what ways do the higher homotopy groups $\pi\_{n}(G)$ with $n>1$ appear in the representation theory of $G$?
As an example, note that there is at least one place where the fundamental group is significant, namely in distinguishing representatio... | https://mathoverflow.net/users/5124 | HIgher Homotopy Groups and Representation Theory | I don't know if this is the sort of thing you are looking for, but the higher *rational* homotopy groups appear naturally in representation theory. For example, if $G$ is a simply connected compact Lie group, then the algebra
$Ext^\ast\_{\mathcal U (\mathfrak g)}(\mathbb C, \mathbb C) = H^\ast(\mathfrak g)$
is a ... | 16 | https://mathoverflow.net/users/7762 | 127785 | 71,186 |
https://mathoverflow.net/questions/127729 | 5 | It is written in Wikipedia <http://en.wikipedia.org/wiki/Groupoid>, that any *connected groupoid* $A\rightrightarrows X$ is isomorphic to an *action groupoid* $G\ltimes X$ coming from a transitive action of some group $G$ on $X$. I do not understand how to construct such a group $G$, and would be grateful for an explan... | https://mathoverflow.net/users/4149 | Connected groupoids and action groupoids | Here's what I wrote on Math Stack Exchange:
A connected groupoid *A* can be written as an action groupoid for many different groups *G*. All the groupoid determines is *H*, the group of automorphisms of any object in the groupoid, and the index of *H* in *G*, which is the cardinality of the set of objects of the grou... | 6 | https://mathoverflow.net/users/644 | 127787 | 71,188 |
https://mathoverflow.net/questions/127796 | 1 | A finite $p$-group is said to be special if $Z(G)=G'$. Is there classification of special $p$-groups? (Please suggest references, if classification is done. If the classification is incomplete, please suggest the references, in which it is done for particular cases. The case $Z(G)=G'\cong \mathbb{Z}/p$ is very well kno... | https://mathoverflow.net/users/33184 | Classification of Special $p$-Groups | The standard definition of a special p-group is more restrictive than that. A $p$-group $G$ is special if either it is elementary abelian, or if $P'=Z(P)=\Phi(P)$ is elementary abelian. So in the second case both $Z(P)$ and $P/Z(P)$ are required to be elementary abelian.
I don't believe that there is any precise clas... | 7 | https://mathoverflow.net/users/35840 | 127803 | 71,194 |
https://mathoverflow.net/questions/127800 | 5 | A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega\_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of course proper) notions of forcing are stationary set preserving.
On the other hand Shelah realized that assuming the Semiproper Forcing Axiom ... | https://mathoverflow.net/users/4753 | Examples of stationary set preserving forcings that are not semiproper? | Namba forcing is stationary preserving but not semiproper unless Chang's Conjecture holds. See "Proper and Improper Forcing" of Shelah, Ch 12.
| 8 | https://mathoverflow.net/users/6647 | 127812 | 71,197 |
https://mathoverflow.net/questions/127799 | 0 | Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and $\phi\_n(x)=\phi(x)+\cdots+\phi(T^{n-1}x)$ be the induced cocycle.
A point $x\in X$ is said to be $\phi$-transient if $|\phi\_n(x)|\to\infty$ as $n\to\inf... | https://mathoverflow.net/users/11028 | Recurrence and transience of cocycle over a dynamical system | I am not quite sure what the question is, but the following result might be a helpful starting point:
Theorem (Giles Atkinson, 1976): Let $T$ be an ergodic invertible measure-preserving transformation of a probability space $(X,\mathcal{F},\mu)$ and let $\phi \colon X \to \mathbb{R}$ be integrable. Then the following... | 2 | https://mathoverflow.net/users/1840 | 127817 | 71,200 |
https://mathoverflow.net/questions/127820 | 1 | Suppose we have a initial segment $x\_1,\ldots,x\_N$ (for reasonably large $N$) of a sequence of natural numbers $(x\_i)$. We have reason to believe the generating function $\sum\_{i=0}^\infty x\_iX^i$ is rational. Are there any methods one could use to guess/approximate this generating function as a quotient of polyno... | https://mathoverflow.net/users/33189 | Approximating rational generating functions | This is called Padé approximation. There are several computer packages that can do that, in particular GFUN (Salvy and Zimmermann) for maple, Guess (Kauers) for mathematica, in FriCAS it's built-in (the function is called guessPade). You can access the latter also from sage, although very likely there is something buil... | 2 | https://mathoverflow.net/users/3032 | 127826 | 71,202 |
https://mathoverflow.net/questions/127829 | 6 | Assume that $(M^n,g)$ is an $n$ dimensional ($n \geq 3$) closed Riemannian manifold with constant scalar curvature and $Ric\_g$ nonnegative. Then is $g$ Einstein?
| https://mathoverflow.net/users/29480 | when constant scalar curvature implies Einstein? | There is no reason for this, and the answer is indeed **no**.
The simplest example I can think of is the product of two $\mathbb{S}^2$, each endowed with round metrics *of different radius* (added in edit). This manifold is homogeneous and thus has constant scalar curvature, its sectional curvature is non-negative so... | 11 | https://mathoverflow.net/users/4961 | 127832 | 71,206 |
https://mathoverflow.net/questions/127809 | 2 | Does anyone have a reference that the Albanese is dual to the Picard scheme (under suitable conditions)?
Edit: In fact, the following is true: $(\mathrm{Pic}^0(X)\_{\mathrm{red}})^\vee = \mathrm{Alb}(X)$, and the Picard scheme is reduced (and then smooth and an Abelian scheme) iff equality holds in $\dim H^1(X,\mathc... | https://mathoverflow.net/users/nan | Albanese dual to the Picard scheme | The nicest modern reference for the theory of the Albanese that I know of is the appendix to [this](http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20I.pdf) article of S. Mochizuki.
| 6 | https://mathoverflow.net/users/259 | 127835 | 71,207 |
https://mathoverflow.net/questions/127823 | 2 | Oftentimes in density arguments we let $\{x\_n\}$ be a dense sequence and this is sufficient to imply the desired result.
From a research question I am working on I have simplified the example/counterexample to the following problem, which I believe is perhaps a nice exercise in choice (and yet, I cannot make a good... | https://mathoverflow.net/users/28090 | Finding a good ordering of $\mathbb{Q}$ | The answer is **no**.
First, the ordering and density hypothesis are irrelevant (you do not use the ordering, and the density can be managed independently of the measure assumption we are trying to satisfy).
The Lebesgue measure of the set of $x\in(-1,1)$ such that $x\in B(x\_,;r\_n)$ for at least one
$n>N$ is at m... | 6 | https://mathoverflow.net/users/4961 | 127836 | 71,208 |
https://mathoverflow.net/questions/126836 | 1 | Has the following generalized version of the Lehmer's conjecture for the Euler totient function
(original version: there are no composite solutions to the equation $n-1 \equiv 0 (\varphi(n))$)
Find composite solutions to the equation
$2(n-1) \equiv 0(\varphi(n))$
has been examined ever before?
| https://mathoverflow.net/users/31236 | Generalized Lehmer Euler Conjecture | I am not an expert here, but I think it has been implicitly considered in the context of $k$-Lehmer numbers.
These are the positive composite integers $n\ge 1$ which satisfy $\phi(n)\mid (n-1)^k$, where
$k\ge 1$ is a fixed positive integer.
There are strong connections between $k$-Lehmer numbers and Carmichael number... | 3 | https://mathoverflow.net/users/32332 | 127852 | 71,213 |
https://mathoverflow.net/questions/127841 | 15 | $\DeclareMathOperator\Top{\mathit{Top}}$I am not sure if its OK to ask this question here.
Let $\Top$ be the category of topological spaces. Let $X,Y$ be objects in $\Top$.
Let $F:\mathbb{I}\rightarrow \Top(X,Y)$ be a function (I will denote the image of $t$ by $F\_t$). Let $F\_{\*}:X\times \mathbb{I}\rightarrow Y$... | https://mathoverflow.net/users/32135 | Giving $\mathit{Top}(X,Y)$ an appropriate topology | Briefly, this works very nicely when $X$ is locally compact, but not otherwise.
Then the function space carries the compact-open topology.
John Isbell gave a survey of the story and literature in his paper
*General Function Spaces, Products and Continuous Lattices*,
in Math Proc Cam Phil Soc **100** (1986) 193--205.
... | 25 | https://mathoverflow.net/users/2733 | 127853 | 71,214 |
https://mathoverflow.net/questions/127845 | 3 | This a question where I have thought quite long about:
The eigenfunctions (or also normal modes) of an dry Euler beam subject to free-free boundary conditions are given by
$$ \frac{\partial^4\psi}{\partial x^4}=\lambda\_k^4\psi\qquad(0\le x\le1)\,,$$
$$\frac{\partial^2\psi}{\partial x^2}=\frac{\partial^3\psi}{\part... | https://mathoverflow.net/users/33191 | Eigenfunctions of fourth-order differential operator | You need to specify $\lambda\_k$ more clearly. More precisely, what is the spectrum of this operator? The equation
$$ \cos x\cosh x =1 $$
seems to have a unique solution $\mu\_k$ on any interval of the form $(k\pi/2, k\pi/2+\pi)$, $k\in\mathbb{Z}$, so I assume the spectrum might be $\mu\_k^4$, $k\in\mathbb{Z}$?!? (... | 3 | https://mathoverflow.net/users/20302 | 127858 | 71,219 |
https://mathoverflow.net/questions/127871 | 0 | Let the distribution function $CDF(X,t)$ of a random variable $X$ be defined as $0$ for $ t <0, \text{Cantor function}(t)$ for $t \ge 0$ and $ t \le 1, 1$ for $ t > 1$ (for example, see <http://en.wikipedia.org/wiki/Cantor_function> for the used definition). What is the characteristic function of $X$? It would be an ex... | https://mathoverflow.net/users/32273 | What is the characteristic function of the devil’s staircase? | This should probably be a comment but I'm 9 points short.
The answer's on wikepedia.
It's $e^{\tfrac{it}2} \prod\_{i=1}^\infty cos\left(\frac t{3i}\right)$.
<http://en.wikipedia.org/wiki/Cantor_distribution>
I'm pretty sure there's no closed form of the infinite product.
| 2 | https://mathoverflow.net/users/32372 | 127875 | 71,225 |
https://mathoverflow.net/questions/127843 | 2 | For a topological space $X$ and a positive integer $k\in \mathbb{N}\_{>0}$ let $F\_k(X):= \{ (x\_1,\ldots,x\_k)\in X^k |x\_i\neq x\_j \text{ for } i\neq j \}$ be its $k$-configuration space.
Let $f:M\to \mathbb{R}$ be a Morse function on a compact manifold $M$.
The space $GVect(f)$ of all gradient-like vector fields f... | https://mathoverflow.net/users/29827 | Contractibility of a configuration space | It seems that $GVect(f)$ is a manifold without boundary.
Building on my [answer](https://mathoverflow.net/questions/123991) to your last question (did you prove all of it?) let us argue as follows:
A vector field $X$ is in
$GVect(f)$ if:
* $X(p)=0$ for each critical point $p$ of $f$. This describes a closed linear ... | 3 | https://mathoverflow.net/users/26935 | 127879 | 71,226 |
https://mathoverflow.net/questions/127739 | 6 | Actually I have a few related questions.
Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}\_2(\mathbb{Z})\backslash\mathcal{H}$.
I know $Y(1)$ is only a coarse moduli space, so there isn't a universal elliptic curve over it, but **does there exist an elliptic surface over it such that the fiber above every poin... | https://mathoverflow.net/users/15242 | Is there an elliptic surface over $Y(1)$? | 1 The last exercise of Hartshorne chapter 4 section 4 proves that there is no elliptic surface over $\mathbb A^1$ with nonconstant $j$ invariant. Obviously this is a special case.
2 No as well. Put the fiber in Weirstrass form near $j=0$. Then one can write it as $y^2=x^3-g\_2x-g\_3$, where $g\_2$ and $g\_3$ are both... | 6 | https://mathoverflow.net/users/18060 | 127890 | 71,232 |
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