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https://mathoverflow.net/questions/57194 | -1 | given two $C^{\*}$ algebras $A\subset B $, acting on the same Hilbert space $H$, and $\delta $ is a derivation from $A$ into $B$,(in this case, a derivation is a linear mapping such that $\delta(ab)=a\delta(b)+\delta(a)b,\forall a,b \in A$) and assume it is bounded, then, is there an element $h$ of $A^{-}$(the weak ope... | https://mathoverflow.net/users/9305 | derivation between two $C^{*}$ algebras | The answer is no. Let $A$ be commutative von Neumann algebra. Let $T\in B(H)$ be such that $Ta-aT\neq 0$ for some $a\in A$. Then $\delta(a)=Ta-aT$ is bounded derivation but there is no element $h\in A^{-}=A$, such that $\delta(a)=ha-ah$ for all $a\in A$.
| 6 | https://mathoverflow.net/users/8699 | 57199 | 35,678 |
https://mathoverflow.net/questions/57197 | 6 | I saw the following statement made without proof in a paper of Bogomolov and Tschinkel:
If $X$ is an algebraic surface, and $C$ is an ample smooth curve in $X,$ then the fundamental group of $C$ surjects onto that of $X.$
I was wondering if someone could supply a reference, and perhaps some idea of what the most ge... | https://mathoverflow.net/users/11142 | fundamental groups of curves | There is a general Lefschetz hyperplane theorem for the homotopy groups of an ample divisor $D$ on a smooth complex variety $X$. Basically, this theorem says that the relative homotopy groups $\pi\_i(X,D)$ are zero for all $i$ less than $\dim X$. In particular, the map $\pi\_1(D)\to\pi\_1(X)$ is an isomorphism for $\di... | 10 | https://mathoverflow.net/users/3996 | 57201 | 35,679 |
https://mathoverflow.net/questions/57109 | 13 | Hi,
I asked this question already on math.stackexchange but got no answer (link: <https://math.stackexchange.com/questions/22155>).
Our setting: An Euclidean vector bundle $(E, h, \nabla^E)$ over a Riemannian manifold (M,g) is said to have bounded geometry, if the norms of the curvature tensor $R^E$ and of all its ... | https://mathoverflow.net/users/13356 | Riemannian manifold of bounded geometry has a normal bundle of bounded geometry | Interesting question. The answer is no: surfaces with bounded geometry can have normal
bundles with unbounded curvature.
To set the stage, it's worth first noting that you can have a surface with extreme geometry isometrically embedded in $\mathbb E^3$, where the normal bundle, being one-dimensional, has a trivial co... | 17 | https://mathoverflow.net/users/9062 | 57208 | 35,684 |
https://mathoverflow.net/questions/57192 | 6 | Is there a solution/approximation for the non-linear difference equation $c\_n = c\_{n-1}+c\_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$?
| https://mathoverflow.net/users/13368 | Is there a solution/approximation for the non-linear difference equation $c_n = c_{n-1}+c_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$? | As Aaron Meyerowitz mentions when $\alpha=\frac{1}{b}$ the sequence is related to the number of partitions of $bn$ into powers of $b$. The asymptotic value of this sequence was determined by de Bruijn (On Mahler's partition problem). I believe his methods can be used to get asymptotic values for general $\alpha$, thoug... | 6 | https://mathoverflow.net/users/2384 | 57210 | 35,686 |
https://mathoverflow.net/questions/57185 | 6 | In terms of matrix theory, the question I'm led to is the following: Start with an $n$-dimensional vector space over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, which has a non-degenerate symmetric bilinear form. Consider all $n \times n$ skew-adjoint matrices $A$ relative to this form.
... | https://mathoverflow.net/users/4231 | Nilpotent matrices related to Lie algebras of special orthogonal groups in characteristic 0 | Let $\lambda$ be a partition of $n.$ Then there exists a skew-symmetric nilpotent matrix whose Jordan blocks sizes are $\lambda\_i$ if and only if every even parts has even multiplicity. This follows easily from the representation theory of $\mathfrak{sl}\_2$ and is duly recorded in standard sources, e.g. Collingwood a... | 4 | https://mathoverflow.net/users/5740 | 57211 | 35,687 |
https://mathoverflow.net/questions/50990 | 10 | Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane.
Observe that if $s=\sigma + it$ with $\sigma>1$ real and finite (or at least infinitesimally close thereto), but $t$ infinite, then the summands $n^s= n^\sigma(\cos \ln(n)t + i\sin \... | https://mathoverflow.net/users/10909 | Non-standard enlargements, $\zeta(s)$ and analytic continuation | I just saw this on arXiv: [*Nonstandard Mathematics and New Zeta and L-Functions*](http://arxiv.org/abs/0808.1965), the PhD thesis of one Benjamin Clare of U. Nottingham.
>
> This Ph.D. thesis, prepared under the supervision of Professor Ivan Fesenko, defines new zeta functions in a nonstandard setting and their an... | 8 | https://mathoverflow.net/users/10946 | 57212 | 35,688 |
https://mathoverflow.net/questions/57158 | 1 | Let $\sigma(x) = \sigma\_1(x)$ denote the sum of all the positive divisors of $x$.
If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following equation?
$$2{n^2}\sigma(n) = \sigma({n^2})\sigma(\sigma(n))$$
In other words, does there exist such an odd $N = {n^2}\si... | https://mathoverflow.net/users/10365 | Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request] | (Thanks to Luis H Gallardo for pointing out the parity condition on $n$.)
(Edited on March 12, 2015)
I was actually trying to (initially) rule out the condition $\sigma(n) = q^k$ for an odd perfect number given in the Eulerian form $N = {q^k}{n^2}$. Just today, I found out that it might be possible to rule out the ... | 1 | https://mathoverflow.net/users/10365 | 57213 | 35,689 |
https://mathoverflow.net/questions/57215 | 5 |
>
> **Possible Duplicate:**
>
> [Cohomology and fundamental classes](https://mathoverflow.net/questions/1489/cohomology-and-fundamental-classes)
>
>
>
Hello,
I would like to know if all homological classes in a smooth manifold can be represented as immersed submanifolds, or examples where this is not true... | https://mathoverflow.net/users/7894 | homology classes as immersed submanifolds | It might be better to split the question into 2 cases and 2 steps.
**Step 1**: Which homology classes in $X$ can be represented by continuous maps of closed smooth manifolds (ie, which classes are $f\_\*[M]$ where $f\colon M\to X$ is a map from a closed manifold with fundamental class $[M]\in H\_\*(M)$)? This is the ... | 19 | https://mathoverflow.net/users/8103 | 57218 | 35,691 |
https://mathoverflow.net/questions/57217 | 8 | If $G$ is a group of square-free order with at-least three prime factors, $|G|=p\_1p\_2....p\_r$, $(2< p\_i < p\_{i+1})$, does $G$ contain a **cyclic subgroup** of composite order?
(As groups of square-free order are solvable, $G$ will necessarily have a proper subgroup of composite order.)
| https://mathoverflow.net/users/12484 | Subgroups of groups of Square-free order | Yes, $G$ always contains a cyclic subgroup of composite order. Note that all Sylow subgroups of $G$ are cyclic, i.e. $G$ is a Zassenhaus metacyclic group. Such groups have a very precise structure: they are of the form
$$ G = \left\langle a, b \mid a^m = b^n = 1, b^{-1} a b = a^r \right\rangle ,$$
where $m,n,r$ satisfy... | 12 | https://mathoverflow.net/users/12858 | 57223 | 35,694 |
https://mathoverflow.net/questions/57224 | 4 | Let us say that a poset $P$ is $\mathbf{\kappa}$**-directed** iff every collection of fewer than $\kappa$-many elements in $P$ has an upper bound in $P$. $P$ has the $\mathbf{\kappa}$ **chain condition** iff every antichain in $P$ has size less than $\kappa$. Consider statements of the form:
>
> Every poset $P$ wit... | https://mathoverflow.net/users/7521 | Decomposing a poset into directed subposets | You can look at: S. Todorcevic, Directed sets and cofinal types.
| 2 | https://mathoverflow.net/users/11094 | 57231 | 35,697 |
https://mathoverflow.net/questions/48209 | 2 | Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form
$$\sum\_{i=1}^{r}{{\alpha\_i}{\log(q\_i)}}$$
where the $$\alpha\_i$$ are positive integers (not necessarily distinct) and the $$q\_i$$ are odd primes?
The reason why I ask this question is because ... | https://mathoverflow.net/users/10365 | Reference Request - Sharp Estimates for a Logarithmic Sum | Rather than multiplying, we sum $\forall i \in {1, 2, \ldots \omega(N)}$ to get:
$$\sum\_{j = 1}^{\omega(N)}\frac{{{q\_j}^{\beta\_j}}{\sigma({q\_j}^{\beta\_j})}}{N} \le \frac{2\omega(N)}{3}$$
Following Nielsen, we know that (for lack of an "effective" upper bound for $\omega(N)$):
$$\inf\left({\frac{2\omega(N)}{3... | 0 | https://mathoverflow.net/users/10365 | 57233 | 35,698 |
https://mathoverflow.net/questions/56826 | 9 | Consider a random walk on a connected graph $G=(V,E)$. That is, associate to each neighbouring nodes $a,b\in V\ $ transition probabilities $\mathbb{P}(a\rightarrow b), \mathbb{P}(b\rightarrow a) $ that define a Marcov process with the state space $V$. Fix one point $\mathcal{O}\in V$ and a set $D\subset V$. Define $\ma... | https://mathoverflow.net/users/11521 | Probability of return vs. probability of return in minimal number of steps | Your question is slightly ambiguous.
I think that there is almost no relation between $P(x), P(y)$ and $P\_d(x),P\_d(y)$.
Here is the reason:
You can take any graph $G(V,E)$ and add two edges to it edge from $x$ to $O$ and edge from $y$ to $O$ such that $Pr(x\mapsto O)=\varepsilon\_1$ and $Pr(y\mapsto O)=\varepsil... | 3 | https://mathoverflow.net/users/4246 | 57237 | 35,700 |
https://mathoverflow.net/questions/57216 | 2 | My textbook claims: P \subset P/Poly, and that this is proper.
It claims that all unary languages are in P/Poly, and then goes on to claim that UHALT = {1^n | n encodes (M,x) s.t. M halts on x } is in P/Poly, but not in P.
I understand the following thing:
```
(1) HALT can not be calculated by any TM
(2) UHALT c... | https://mathoverflow.net/users/3609 | Why is UHALT in P/Poly? | Fix $n$, we want to construct a circuit $C\_n$ deciding UHALT on inputs $w$ of length $n$. Now, either $n\notin\mathrm{HALT}$, in which case $C\_n$ is the constant $0$ circuit, or $n\in\mathrm{HALT}$, in which case $C\_n$ is
$$C\_n(w)=\begin{cases}1&\text{if }w\text{ is a string of 1s,}\\0&\text{otherwise,}\end{cases... | 3 | https://mathoverflow.net/users/12705 | 57242 | 35,704 |
https://mathoverflow.net/questions/57245 | 5 | Suppose we have a $n\times m$ rectangular grid (namely: $nm$ points disposed as a matrix with $n$ rows and $m$ columns).
We randomly pick $h$ different points in the grid, where every point is equally likely.
If only horizontal or vertical movements between two points are allowed, what is the probability that the p... | https://mathoverflow.net/users/13383 | Random points in a rectangular grid defining a closed path | Having screwed up the answer by getting the wrong answer for a really simple calculation the first time, I'm now going to try to redeem myself.
First, to make things easier, let each point be present with probability $h/(mn)$, and let these random variables be independent. This won't change the probabilities much.
... | 10 | https://mathoverflow.net/users/2294 | 57249 | 35,707 |
https://mathoverflow.net/questions/57235 | 15 | What is the group of outer automorphisms of $SL\_n(\mathbb{Z})$. I wanted to understand semidirect products of the form $SL\_n(\mathbb{Z})\rtimes\_\varphi \mathbb{Z}$ and its isomorphism type depends only on $[\varphi]\in Out(SL\_n(\mathbb{Z})$. There is always the conjugate inverse, which is clearly not an inner autom... | https://mathoverflow.net/users/3969 | Automorphisms of $SL_n(\mathbb{Z})$ | As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out the... | 9 | https://mathoverflow.net/users/4231 | 57254 | 35,709 |
https://mathoverflow.net/questions/57230 | 25 | Consider the Möbius function $\mu (m)$. (Thus $\mu(m)=0$ unless all prime factors of $m$ appear once and $\mu (m)=(-1)^r$ if $m$ has $r$ distinct prime factors.) Next consider for some natural number $n$ the discrete Fourier transform
$$\hat \mu (k)= \frac1n \sum \_{r=0}^{n-1} \mu(r)e^{2 \pi i rk/n}.$$
So $\sum \_{... | https://mathoverflow.net/users/1532 | Discrete Fourier Transform of the Möbius Function | Here is an answer to 1. It is known that for any $A > 0$ that $\sum\_{m \leq x} \mu(m) e(\alpha m) = O\_A(x (\log{x})^{-A})$ uniformly in $\alpha$. For instance, consult Theorem 13.10 of Iwaniec and Kowalski's book, Analytic Number Theory. This uniform bound comes from combining the zero free region of Dirichlet L-func... | 16 | https://mathoverflow.net/users/2627 | 57257 | 35,710 |
https://mathoverflow.net/questions/57246 | 3 | I am finishing up a paper and I would like to be able to quote a theorem that does what
is said in the title. To be specific let me introduce some notations:
${\bf F}$ is a local field of charateristic $p>0$, $|\cdot|$ is the absolute value on ${\bf F}$ normalized by requiring that $|\pi|=q^{-1}$ where $\pi$ is a unifo... | https://mathoverflow.net/users/11018 | Reference request - spectral radius formula for linear transformations in char p | I don't know of a reference off hand, but I don't see any harm in just claiming that the assertion easily follows from the Jordan decomposition of $T$.
After all, if you know that the limit on the left hand side exists you just write $T = D + N$ for some diagonalizable over $\mathbf{F}$ endomorphism $D$, and some nil... | 1 | https://mathoverflow.net/users/6827 | 57260 | 35,712 |
https://mathoverflow.net/questions/57261 | 0 | Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?
| https://mathoverflow.net/users/13387 | Small set of acts over a countable monoid? | I assume that $S$-act means a set with an $S$-action that can't be decomposed into two nonempty subsets both invariant under $S$. Then the answer to the question is no. Let $S$ be the 2-element monoid that isn't a group, i.e., the multiplicative monoid $\{0,1\}$, and let $\kappa$ be any non-zero cardinal number. Then t... | 1 | https://mathoverflow.net/users/6794 | 57262 | 35,713 |
https://mathoverflow.net/questions/44734 | 3 | This question is about conditions on a mother wavelet that generates a countable familily of child wavelets via scaling and translation, that are both necessary and sufficient for the child wavelets to form a frame in the Hilbert space $L^2(\mathbb{R})$
Here are the precise definitions of these concepts in the contex... | https://mathoverflow.net/users/1478 | When does a mother wavelet generate a frame? | No. This question is, I believe, open even for functions of the type $\hat \psi = I\_E$, dilations by powers of 2 and translation by integers. See, for example the relatively recent paper of Bownik and Weber <http://pages.uoregon.edu/mbownik/papers/16.pdf>, where specific examples of $\psi$ of this type are given for w... | 3 | https://mathoverflow.net/users/13386 | 57263 | 35,714 |
https://mathoverflow.net/questions/57141 | 9 | Let $X$ be a compact complex Kahler manifold with first real Chern class $c\_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X\_s$ are compact Kahler with $c\_1(X\_s) = 0$ for $s \not= 0$. Do we know that the central fiber $X\_0$ is Kahler with $c\_1 = 0$?
... | https://mathoverflow.net/users/4054 | Is the deformation limit of Ricci-flat Kahler manifolds Kahler? | There are counterexamples: a Moishezon manifold, which has trivial canonical class and is birationally equivalent to a hyperkahler manifold,
is also deformationally equivalent to a hyperkaehler manifold (this is a result
of Huybrechts, I am not sure if he stated it in this generality, but his
proof certainly works). Su... | 12 | https://mathoverflow.net/users/3377 | 57270 | 35,718 |
https://mathoverflow.net/questions/57256 | 4 | Let $X$ be a smooth algebraic variety, $A \subset X$ a smooth subvariety, $f:Y \to X$ the blow-up of $X$ along $A$ and $M$ a quasi-coherent $O\_X$-module (in the case I'm interested in, $M$ is actually a $D\_X$-module but it doesn't change a thing).
**Question:** Is there a natural resolution of $O\_Y$ as a $f^{-1}O... | https://mathoverflow.net/users/1985 | Pull-back on a blow-up | The easiest way to compute is the following. Assume that $A$ is the zero locus of a regular section $s$ of a vector bundle $E$ on $X$ (this is always the case locally). Then there is an embedding $i:Y \to P\_X(E)$ over $X$. Then $i\_\*O\_Y$ has a nice resolution.
Indeed, consider the relative Grothendieck line bundl... | 8 | https://mathoverflow.net/users/4428 | 57275 | 35,721 |
https://mathoverflow.net/questions/57273 | 1 | In the usual (fomal) construction of the quantum general linear group $GL\_q(N)$, an Ore extension is used. See for example Kassel. Why is this necessary? Surely one can just augment the set of generators with an element det$^{-1}$ and the set of relations with the relations det.det$^{-1} = 1$ and det$^{-1}$.det$=1$ an... | https://mathoverflow.net/users/12653 | Ore Extensions and the Construction of the Quantum General Linear Group | I have hard time understanding what is your notion of usual; most references indeed define the quantum linear group $M\_q(N)$ by generators and relations (or by the universal property as defined by Manin) and then localize at $det\_q$. So far as the construction goes. But it is of course very useful to know that one ca... | 4 | https://mathoverflow.net/users/35833 | 57278 | 35,722 |
https://mathoverflow.net/questions/57225 | 11 | Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, Kunen is always very careful with letting this C.T.M. to model $only$ finite fragments of ZFC. I recently read in one of th... | https://mathoverflow.net/users/13059 | Kunen's use of Countable Transitive Models | Here are some examples that might help in understanding. If ZFC is consistent, then it follows we have a *set* model $M$ of the theory. Consider a nonprincipal ultrafilter $U$ on $\omega$ and let $M^{\omega}/U$ be the induced ultrapower. $M^{\omega}/U$ is a model of ZFC, but it cannot be well-founded because its $\omeg... | 6 | https://mathoverflow.net/users/11318 | 57285 | 35,724 |
https://mathoverflow.net/questions/57277 | 6 | Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^\*$ is a surjective isometry of $X^\*$.
In general, a linear map $g:X^\* \to X^\*$ need not have a pre-adjoint. But what if $g$ is a surjective isometry? Must there exist $f:X \to X$ such tha... | https://mathoverflow.net/users/13391 | Must a surjective isometry on a dual space have a pre-adjoint? | Let $X$ be the space of sequences, indexed by the nonzero integers, that tend to $0$ at $-\infty$ and to an arbitrary finite limit at $\infty$, with sup norm (a direct product of $c\_0$ and $c$). Then $X^\*$ can be identified with $\ell^1$ of $\mathbb{Z}$. If $f$ is in $\ell^1$, then the corresponding functional on $X$... | 6 | https://mathoverflow.net/users/1119 | 57286 | 35,725 |
https://mathoverflow.net/questions/57280 | 0 | I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] cross [c,d],
the number is 1 if k is in the interior
the number is 2 if k is on an edge
the number is 4 if k i... | https://mathoverflow.net/users/6137 | Numbers associated with boundaries of manifolds | I was calculating the approximate density of a set A (line segment, square, cube, etc) in a ε-neighbourhood of a point x. It's related to the Lebesgue's density theorem. (Actually I was calculating the inverse of these numbers.)
| 0 | https://mathoverflow.net/users/6137 | 57287 | 35,726 |
https://mathoverflow.net/questions/57269 | 7 | This likely isn't a research-level question, but it is at least a question of interest to this researcher. I'm happy with an answer that sends me somewhere (preferably online) to read about a well-known (to somebody) solution, if there is such a thing. First the question, then the motivation.
>
> Let $X\_1,X\_2,\do... | https://mathoverflow.net/users/935 | How to estimate a time distribution | As requested, I'm making my comment an answer.
While this does not answer the problem as you've formalized it, I would argue that the correct method to consider this problem is actually Weibull analysis--generally failure times are not distributed normally. Essentially, plot the failure times on Weibull paper; the sl... | 3 | https://mathoverflow.net/users/6950 | 57289 | 35,727 |
https://mathoverflow.net/questions/57239 | 4 | Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a symmetric bilinear form and write
$$
A = \sum\_{k\ge 1} \lambda\_k \varphi\_k\otimes \varphi\_k \tag{1}
$$
My question is:
... | https://mathoverflow.net/users/8146 | An analogue of Hilbert-Schmidt theorem for multilinear forms | Not in general. For even $n=2m$, $m >1$, a $n$-linear symmetric form $A$ is a bilinear
symmetric form (or operator) on $H^{\otimes\_s m}\otimes H^{\otimes\_s m}$ and it will have a spectral decomposition of the form $$A=\sum\_k \lambda\_k u\_k \otimes u\_k$$ with $u\_k \in H^{\otimes\_s m}$, which is not in general in ... | 3 | https://mathoverflow.net/users/12409 | 57300 | 35,731 |
https://mathoverflow.net/questions/57295 | 11 | When I teach calculus, I really try to stress the importance of knowing the domain of a function.
One example that I sometimes like to use to show students the importance of inspecting the domain is the following: $f(x) = \sqrt{\sin(x)-1}$. I ask the student to differentiate this function, and they happily apply the... | https://mathoverflow.net/users/1106 | Can formally differentiating give a derivative of a discrete function? | How about $y=\log(-x^2)$?
| 13 | https://mathoverflow.net/users/3684 | 57303 | 35,733 |
https://mathoverflow.net/questions/57297 | 8 | The relations $R$ in abstract graphs (with genuinely propertyless vertices) cannot be *defined* because there is nothing the relations can base on: they have to be presupposed.
But consider derived relations $\Phi(x,y)$ between vertices of a graph which can be defined in terms of the base relation $R$. I don't want t... | https://mathoverflow.net/users/2672 | Self-defining structures | I've found a necessary and sufficient characterization for
when a relation $\Phi$ is nontrivially self-fulfilling, in
the theorem below.
(As Aaron pointed out, every $\Phi$ is realized trivially
in the graph with no vertices and also in the graph with
one vertex, so by *nontrivially* self-fullfilling, let us
insist t... | 4 | https://mathoverflow.net/users/1946 | 57306 | 35,734 |
https://mathoverflow.net/questions/57304 | 9 | Let $X$ be a space. The symmetric group $\Sigma\_{n+1}$ acts on the function space
$$
X^{\Delta^n}
$$
of continuous maps from the standard $n$-simplex to $X$. The action is induced
by permuting the vertices.
Let $\Sigma\_{n+1}$ act on $\Bbb Z$ by means of the sign representation.
Then the singular $n$-cochains
$$... | https://mathoverflow.net/users/8032 | "Skew Cohomology" of a Space | I believe it should be exactly the same as ordinary singular cohomology. It should define a cohomology theory for the exact same reason that usual singular cohomology does (the usual proof of excision by subdivision seems to work since the cosubdivision of a $\Sigma\_n$-invariant cochain can be checked to still be $\Si... | 10 | https://mathoverflow.net/users/75 | 57309 | 35,736 |
https://mathoverflow.net/questions/57315 | 7 | In p-adic mathematics, what is the advantage of using Teichmuller representatives over using just the numbers 0,1,2,...,p-1 ?
In either case, the norm is the same.
In either case, all the points are in the center.
Is is fair to say: "Teichmuller representives, just like the digits 0,1,2,...,p-1 , are just names of ... | https://mathoverflow.net/users/13397 | Why use Teichmuller representatives? | The Teichmüller lift can be seen as a map of multiplicative monoids from $\mathbb{F}\_p$ to $\mathbb{Z}\_p$, that is the unique multiplicative section of the mod $p$ reduction map. In particular, the lift produces roots of unity from nonzero inputs. You cannot say that about the integers from $1$ to $p-1$ (which tend t... | 17 | https://mathoverflow.net/users/121 | 57321 | 35,741 |
https://mathoverflow.net/questions/57338 | 1 | Hi,
I'm trying to study a list of vectors. The vectors represent particular states of an organic system over time. The vectors grow in size (relatively fast) and (absolute) value of entries (relatively slowly) over time.
A sequence of these vectors displays the characteristic that adjacent vectors are quite similar... | https://mathoverflow.net/users/13403 | FAIL: First Principles Differentiation to Discover Function / Pattern, '$d_nF$' grows with n | Just some notes (I am not an expert on this, and a different unknown):
Phenomena like this are not unique to a discrete model.
Consider a function, $f$ on say $(c,1)$ with $c>0$ but small whose first derivative is
$x sin (1/x)$ this is small for small $x$.
But, if you differentiate again you get $sin(1/x) - ... | 2 | https://mathoverflow.net/users/nan | 57341 | 35,752 |
https://mathoverflow.net/questions/57323 | 9 | Let $A$, $B$, and $C$ be commutative rings such that $A\otimes\_C B$ makes sense. If $W\_n(A\otimes\_C B), W\_n(A), W\_n(C),$ and $W\_n(B)$ are the length $n$ Witt vectors of the rings $A,B,C,$ and $A\otimes\_C B$. Is it true that
$$
W\_n(A\otimes\_C B)\cong W\_n(A)\otimes\_{W\_n(C)}W\_n(B)?
$$
It seems as though ... | https://mathoverflow.net/users/6310 | Tensor product of rings of Witt vectors | When $B$ is étale over $C$ and $A$ or $B$ is finite over $C$,
then the result is known by Theorem (2.4) in my paper
[Descent for the $K$-theory of polynomial rings](http://www.digizeitschriften.de/id/266833020_0191%7Clog39?tify=%7B%22pages%22:%5B412,413%5D,%22view%22:%22%22%7D&origin=/search?q%3Ddescent%2520for%2520the... | 14 | https://mathoverflow.net/users/4794 | 57349 | 35,757 |
https://mathoverflow.net/questions/57325 | 1 | Let $S$ be a $K3$ surface and $H$ an ample line bundle on it. Fix a Mukai vector $v\in H^\*(S,\mathbb Z)$. If $v$ is primitive, it turns out that Gieseker stability w.r.t.$H$ coincides with Gieseker semistability w.r.t. $H$. Does the same hold for $\mu\_H$-stability, too?
| https://mathoverflow.net/users/33841 | When do stability and semistability coincide? | If the rank and degree (wrt $H$) are coprime, then $\mu$-stability and $\mu$-semistability coincide. The argument is the same as for Gieseker-stability, but simpler.
| 2 | https://mathoverflow.net/users/4716 | 57357 | 35,763 |
https://mathoverflow.net/questions/57291 | 1 | If we have a variety X , over a field k, and x is a geometric point of X, and let $\bar x $be a geometric point of $X\_{k^s} := X \times\_k k^s $above x then we have the following short exact sequence:
$1 \rightarrow \pi\_1(X\_{k^s}, \bar x) \rightarrow \pi\_1(X,x) \rightarrow Gal(k) \rightarrow 1$
Implicit in this... | https://mathoverflow.net/users/3945 | Image of spliting of short exact sequence of algebraic fundamental groups | First a remark : you have to suppose $X$ geometrically connected to get this exact sequence.
I don't know what kind of understanding you expect, but the answer to the question will it always be the full group $\pi\_1(X,x)$ is certainly no. The reason is that there are explicit examples (of proper smooth curves over a n... | 2 | https://mathoverflow.net/users/11682 | 57363 | 35,768 |
https://mathoverflow.net/questions/57388 | 1 | Suppose that, for every Hilbert space $H$, we have a subset $I(H) \subseteq B(H)$ of bounded linear operators on $H$, and that together all $I(H)$ form a two-sided ideal, in the sense that whenever $h \in I(H)$, also $f \circ h \circ g \in I(K)$ for any bounded linear maps $f \colon H \to K$ and $g \colon K \to H$. To ... | https://mathoverflow.net/users/10368 | Are these ideals in rings of operators on Hilbert space unique? | No proper ideal satisfies your condition. Take the Hilbert space to be $ \ell\_2\oplus \ell\_2$ and define $f(x,y)=(o,x)$; $g(x,y)= (x,0)$.
| 2 | https://mathoverflow.net/users/2554 | 57393 | 35,786 |
https://mathoverflow.net/questions/53744 | 1 | Jones (1985) defines a simplified trace invariant for knots by $W\_K(t)=\frac{1-V\_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W\_K(i)$. Does this work for oriented links as well? If not, is there a variant that does?
| https://mathoverflow.net/users/12606 | Simplified Jones trace invariant for links | Oops, my bad. Since the Arf invariant is not defined for all links, what I was looking for doesn't exist. If the Jones polynomial of an oriented link $L$ in $S^3$ equals 0 when evaluated at $t=\imath$, then the Arf invariant of $L$ is not defined. (See W.B. Likorish, \textit{An Introduction to Knot Theory}, p. 106.)
| 2 | https://mathoverflow.net/users/12606 | 57401 | 35,791 |
https://mathoverflow.net/questions/57370 | 18 | This question is prompted by [this one](https://mathoverflow.net/questions/57304/skew-cohomology-of-a-space) (and some of the comments that it drew).
Simplicial complex is to ordered simplicial complex as $X$ is to simplicial set. The question is about $X$.
Let $\text{Unord}$ be the full subcategory of $\text{Set}$... | https://mathoverflow.net/users/6666 | Analogue of simplicial sets | I have never thought of a counter example, but I would not bet on a positive answer to the first question. However, the answer to the second question is yes: this version of the singular functor is a right Quillen equivalence for a suitable model category on $\mathrm{Set}^{\mathrm{Unord}^{op}}$.
As mentioned above by... | 13 | https://mathoverflow.net/users/1017 | 57409 | 35,795 |
https://mathoverflow.net/questions/57405 | 3 | Let $X$ be a one-dimensional Noetherian scheme over an algebraically closed field $k$. Suppose $X$ is reduced and let $X=\bigcup X\_i$ be the composition of $X$ into irreducible components. Then, is the following homomorphism surjective?
$\mathrm{Pic} X\to \bigoplus \mathrm{Pic} X\_i$.
| https://mathoverflow.net/users/13418 | Surjectivity of a homomorphism between Picard groups | Yes! This can be shown using the isomorphism $H^1(X,\mathcal{O}\_X) \cong \mathrm{Pic} X$. First, look at the short exact sequence:
$ 0 \to \mathcal{O}\_X^\* \to \bigoplus \mathcal{O}\_{X\_i}^\* \to \mathcal{C} \to 0$
From the long exact sequence of cohomological groups associated to the short exact sequence, it su... | 4 | https://mathoverflow.net/users/9035 | 57411 | 35,796 |
https://mathoverflow.net/questions/57402 | 10 | (This is a follow-up question to [this](https://mathoverflow.net/questions/56563/why-does-homotopy-behave-well-with-respect-to-fibrations-and-homology-with-respec) one).
As it is nicely outlined in an answer to [this](https://mathoverflow.net/questions/56563/why-does-homotopy-behave-well-with-respect-to-fibrations-an... | https://mathoverflow.net/users/2625 | Analogue to Serre spectral sequence for cofiber sequences and homotopy | There is, sort of, and the idea was developed in the paper "Induced fibrations and cofibrations" by Tudor Ganea in a 1967 paper appearing in the transactions.
The idea is roughly this:
given a cofibration
$$
A \to X \to C ,
$$
let us set $X\_1 = \text{hofiber}(X \to C)$, where the latter means the homotopy fiber.
Th... | 9 | https://mathoverflow.net/users/8032 | 57415 | 35,798 |
https://mathoverflow.net/questions/57386 | 4 | How smooth is the first derivative (in the distribution sense) of a Lipschitz function? Taking difference quotients and testing against an $L^1$ function, we see that $Df$ is in $L^\infty$. In ${\mathbb R}^1$ the converse is true, thanks to the persistence of the formula
$f(x+h) - f(x) = \int\_0^1 f'(x+th) dt~ h$
(... | https://mathoverflow.net/users/7193 | Is the derivative of a Lipschitz function better than L^\infty | Lipschitz functions are exactly $W^{1,\infty}$ (See '[Sobolev space](http://en.wikipedia.org/wiki/Sobolev_space)' on wikipedia - under other examples and perhaps the bit about absolute continuity on lines). This means the short answer to your question is no.
| 6 | https://mathoverflow.net/users/4281 | 57418 | 35,800 |
https://mathoverflow.net/questions/57360 | 2 | Let $I$-identity operator, $\Pi\_N$ is the orthogonal projection in $L\_2$ onto subspace by the first $N$ eigenfunctions of the Stokes operator in $\Omega$, $\alpha\_j$ denotes the increasing sequence of the eigenvalues for the Stokes operator, $c>0$ is the some constant not depending on $N$. We know that $\Pi\_N v = 0... | https://mathoverflow.net/users/13409 | Constant in Poincaré Inequality | This is a fairly standard stuff. Suppose that the Stokes operator $A=-\Delta$ is defined on smooth divergence-free vector fields $u$ which satisfy the standard no-slip boundary condition $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.u\right|\_{\partial\Omega}=0.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad... | 2 | https://mathoverflow.net/users/5371 | 57424 | 35,802 |
https://mathoverflow.net/questions/57410 | 7 | How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis heuristic which will predict what the rate will be?
For example, it is true that
$$ ||u - u^{(\epsilon)}||\_{L^3} \le... | https://mathoverflow.net/users/7193 | Rate of convergence of smooth mollifiers | You can do something with simple scaling as long as you work on the full space. Assume that $|\cdot|$ and $\|\cdot\|$ are two shift-invariant (semi)norms and for the scaling operator $T\_a f(x)=f(x/a)$ you have $|T\_af|=a^t|f|$, $\|T\_a f\|=a^s\|f\|$. Then, if you do the $\delta$-mollifying on $f$, it is equivalent to ... | 8 | https://mathoverflow.net/users/1131 | 57427 | 35,803 |
https://mathoverflow.net/questions/57437 | 10 | Question:
Let $\mathcal{A}$ be an abelian category and $D^?(\mathcal{A})$ be its derived category, where ? could be empty, +, - or b (for boundedness). Is it possible to recover the homological dimension of $\mathcal{A}$ from the derived category?
Here I'm using the term homological dimension in the sense of Gelfan... | https://mathoverflow.net/users/1657 | Does the derived category remember the homological dimension? | Let $V$ be a finite-dimensional vector space, $\mathcal{A}$ be the abelian category of finitely generated graded modules over the symmetric algebra $S(V)$, and $\mathcal{B}$ be the abelian category of finitely generated graded modules over the exterior algebra $\Lambda(V^\*)$. Then the bounded derived categories $\math... | 33 | https://mathoverflow.net/users/2106 | 57445 | 35,811 |
https://mathoverflow.net/questions/57438 | 6 | A morphism of curves is said to be Galois if the corresponding extension of function fields is Galois.
Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective connected curves over $\mathbf{C}$ (or Riemann surfaces if you prefer) with branch locus $B\subset Y$ and ramification locus $R\subset X$. There e... | https://mathoverflow.net/users/4333 | Branch locus of the Galois closure of a Belyi morphism | This is to address the last part of your question -- about Belyi morphism. The answer is YES: you can always get $g: W\to X$ such that $f\circ g: W\to \mathbb CP^1$ is Belyi.
Indeed, $X\to \mathbb CP^1$ is Belyi if it ramifies only over $0,1,\infty$. Let us now conisder $\hat X=f^{-1}(\mathbb CP^1\setminus (0,1,\inft... | 3 | https://mathoverflow.net/users/943 | 57453 | 35,816 |
https://mathoverflow.net/questions/57454 | 5 | Quoting from <http://en.wikipedia.org/wiki/End_(topology)>:
"Let X be a topological space, and suppose that
K1 ⊂ K2 ⊂ K3 ⊂ · · ·
is an ascending sequence of compact subsets of X whose interiors cover X. Then X has one end for every sequence
U1 ⊃ U2 ⊃ U3 ⊃ · · ·
where each Un is a connected component of X \ Kn... | https://mathoverflow.net/users/13434 | Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets? | The main point is this. Let $(L\_k)\_{k=0}^\infty$ be another increasing sequence of compact sets whose interiors cover $X$. Each $X\_n$ is compact and contained in the union of the sets $\text{int}(L\_k)$, so it is contained in some finite union of these open sets. As the sets $\text{int}(L\_k)$ are nested, it follows... | 5 | https://mathoverflow.net/users/10366 | 57458 | 35,819 |
https://mathoverflow.net/questions/57475 | 9 | Is V is a set with n elements, how many different simple, undirected graphs are there with vertex set V?
| https://mathoverflow.net/users/13437 | How many different possible simple graphs are there with vertex set V of n elements | If you consider isomorphic graphs different, then obviously the answer is $2^{n\choose 2}$. Most graphs have no nontrivial automorphisms, so up to isomorphism the number of different graphs is asymptotically $2^{n\choose 2}/n!$. This goes back to a famous method of Pólya (1937), see [this paper](https://www.ncbi.nlm.ni... | 13 | https://mathoverflow.net/users/11919 | 57478 | 35,829 |
https://mathoverflow.net/questions/57392 | 10 | suppose we are given a complete, non-compact Riemannian manifold $(M,g)$. Is it possible to embed (or just immerse) it isometrically into some $R^N$, such that the second fundamental form is bounded? Maybe under some additional assumptions on our manifold and/or metric on it?
This question is a follow-up question to ... | https://mathoverflow.net/users/13356 | Existence of an isometric embedding into Euclidean space with bounded second fundamental form | The curvature tensor can be expressed in terms of second fundamental form.
Therefore bounded curvature is a necessary condition.
Yet injectivity radius has to be bounded below.
These two conditions might be sufficient.
If we assume a bit better regularity (say a bound on covariant derivatives of the curvature tenso... | 10 | https://mathoverflow.net/users/1441 | 57487 | 35,834 |
https://mathoverflow.net/questions/57490 | 1 | From Cantor we know that |R| = 2^|Z|. That is, |R| is equal to the number of subsets of Z. Is it also true that |R| is equal to the number of *infinite* subsets of Z?
| https://mathoverflow.net/users/9714 | Is the number of infinite subsets of Z equal to the size of R? | Yes, because there are only countably many finite subsets of Z.
| 2 | https://mathoverflow.net/users/11919 | 57491 | 35,837 |
https://mathoverflow.net/questions/57493 | 5 | Is there an infinite (edit: but finitely generated) group G such that for all g in G, |g| is finite?
If so, how many such groups exist?
| https://mathoverflow.net/users/9714 | Is there an infinite group whose elements all have finite order? | Yes, as per Ryan's comment you can just take an infinite direct sum of finite groups. However the more interesting problem is: are there (infinite) $\textit{finitely generated}$ groups with all elements of finite order?
The answer to this was open for a long time, but it is indeed yes. In fact this was known as Burn... | 23 | https://mathoverflow.net/users/5732 | 57496 | 35,841 |
https://mathoverflow.net/questions/57351 | 11 | Let $A$ be a unital $C^{\*}$-algebra and $\phi:A \rightarrow A$ be a completely positive map, i.e. $\phi^{(n)}:M\_{n}(A) \rightarrow M\_{n}(A)$ preserves positivity for any natural number $n$, where $\phi^{(n)}((A\_{ij})\_{ij})=(\phi(A\_{ij}))\_{ij}$. It is well-known that the norm $\|\phi\|=sup\_{\|x\|\leq 1}\{\|\phi(... | https://mathoverflow.net/users/6269 | Completely positive maps as "positive operators" | This answer deals with the case that $\phi$ is non-unital. In this case, $\phi$
must be of the form $\phi(a)=ha$, where $h$ is a positive element in the center
of $A$.
Unfortunately, the solution I've got is somewhat long (hopefully correct): Let us assume that $\phi$ is contractive (otherwise one can rescale). Sinc... | 7 | https://mathoverflow.net/users/13381 | 57507 | 35,847 |
https://mathoverflow.net/questions/57436 | 0 | A good day to everyone.
Consider the following "Conjecture":
If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if $\sigma(a) < \sigma(b)$.
I have two questions at this point:
(1) What properties should elements of the set $M$ have to satisf... | https://mathoverflow.net/users/10365 | A simple question regarding the sum-of-divisors function | I think this is a ludicrously broad question. For (1), you could take $M$ to be the set of "champions" for the divisor function, $M=\lbrace1,2,3,4,6,8,10,12,16,18,20,24,\dots\rbrace$, which is <http://oeis.org/A002093> - no conditions on $a$, $b$ necessary. At the other extreme, you could take $M$ to be the whole of th... | 3 | https://mathoverflow.net/users/3684 | 57514 | 35,852 |
https://mathoverflow.net/questions/57515 | 21 | I was trying to understand completely the [post of Terrence Tao on Ax-Grothendieck theorem](https://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/). This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb C^n$ is bije... | https://mathoverflow.net/users/13441 | A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite | $\DeclareMathOperator\Spec{Spec}$To prove Nullstellensatz over $\mathbb{Z}$: as the morphism $f: \Spec(R)\to\Spec(\mathbb Z)$ is of finite type, a theorem of Chevalley says that the image of any constructible subset is constructible. So the image of any closed point by $f$ is a point which is a constructible subset. Th... | 25 | https://mathoverflow.net/users/3485 | 57518 | 35,854 |
https://mathoverflow.net/questions/57426 | 27 | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be s.t. $f(x+y) = f(x) + f(y), \ \forall x, y$
It is quite obvious that this implies $f(cx)=cx$ for all $c \in \mathbb{Z}$ and even further: $\forall c \in \mathbb{Q}$
It is also known that using an infinite dimensional basis for $\mathbb{R}$ over $\mathbb{Q}$, it is possi... | https://mathoverflow.net/users/12597 | Are there any non-linear solutions of Cauchy's equation $f(x+y)=f(x)+f(y)$ without assuming the Axiom of Choice? | It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable.
(This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this wr... | 53 | https://mathoverflow.net/users/6085 | 57532 | 35,859 |
https://mathoverflow.net/questions/57531 | 3 | The following questions should be very easy, but I need help. Notations are same as Bredon's geometry and topology book. This question is related to chapter VII.3.page 441.
$\nabla \colon X\vee X\rightarrow X$ denotes the codiagonal, $S$ denotes the reduced suspension operation.
Q1:Is every example of H-cogroups gi... | https://mathoverflow.net/users/9458 | Examples of H-cogroups and a question about julia sets | Regarding question (1), the answer is that there are co-groups which aren't suspensions.
Berstein, Israel; Harper, John R.
Cogroups which are not suspensions. Algebraic topology (Arcata, CA, 1986), 63–86,
Lecture Notes in Math., 1370, Springer, Berlin, 1989.
A cogroup is a co-H-space with an associative comultipl... | 1 | https://mathoverflow.net/users/8032 | 57534 | 35,861 |
https://mathoverflow.net/questions/57508 | 5 | While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question:
Is it possible to prove that $\mathbb A^2-point$ is not an affine variety, if you don't know that the polynomial ring is... | https://mathoverflow.net/users/13441 | Unique factorisation and the fact that $\mathbb A^2-0$ is not an affine variety? | Yes, you can do it over any field.
First, it is enough to show $\mathcal O(Y) = k[x,y]$ ($Y=A^2-0$). If that is true and $Y$ is affine, then the embedding $Y \to A^2$ must correspond to some $k$-algebra map $k[x,y] \to k[x,y]$, which is absurd.
The key point now, as in Guillermo's post, is to show that $R\_{(x)} \c... | 10 | https://mathoverflow.net/users/2083 | 57547 | 35,868 |
https://mathoverflow.net/questions/57535 | 4 | Hi,everyone.Can someone give me some examples of non-Kahler surfaces whose complex structure and metric structure are all clear?
| https://mathoverflow.net/users/11850 | Examples of non-Kahler surfaces with explicit non-Kahler metric | If your surface is fairly explicit you can obtain an explicit hermitian metric on it as well. For example, if we take Francesco's Hopf surface $X$, then a hermitian metric $\omega$ on $X$ can be identified with a $G$-invariant hermitian metric on $\mathbb C^2 \setminus \{0\}$.
To simplify, let's look at the special c... | 4 | https://mathoverflow.net/users/4054 | 57557 | 35,874 |
https://mathoverflow.net/questions/57511 | 3 | Suppose that $Y$ is a subvariety of $X$, and both are nonsingular. I'm interested in knowing a nice condition to make the inclusion $i\colon Y\to X$ induce an isomorphism of Chow rings $A^{.}(X)\cong A^{.}(Y)$. For example, if $X$ and $Y$ were spaces and I wanted an isomorphism in cohomology, I might hope to show that ... | https://mathoverflow.net/users/5045 | when does an embedding induce an isomorphism of Chow rings? | Hi Graham,
I can't think of too many interesting situations where this happens,
but here is one. If $i:Y\to X$
is the inclusion of the zero section of an algebraic vector bundle, then restriction gives an isomorphism of Chow groups $CH^\*(X)\cong CH^\*(Y)$ (indexed by codimension). A proof can be found in [Fulton, c... | 11 | https://mathoverflow.net/users/4144 | 57563 | 35,877 |
https://mathoverflow.net/questions/57579 | 9 | I'm not a mathematician (I'm an economist) but I hope that this problem is sufficiently non-trivial that someone here will find it interesting.
**Motivation:**
I'm trying to model how workers decide what "skills" to acquire when (a) they have different innate abilities for different skills but (b) they face competi... | https://mathoverflow.net/users/13456 | Existence of a sink in directed graphs with a certain structure | Your conjectures are proven in the paper "Congestion Games with Player-Specific Payoff Functions" by I. Milchtaich, published in Games and Economic Behavior in 1996.
Usually, the term "congestion game" means a game in which players choose nonempty subsets of the set of resources. Each resource is assigned yields a ce... | 12 | https://mathoverflow.net/users/5963 | 57587 | 35,891 |
https://mathoverflow.net/questions/57586 | 8 | So let $X$ be a finite CW complex which is connected.
Q1: Is $\pi\_1(X)$ necssarily a finitely presented group?
If the answer is yes, then how does prove it. I've tried to prove it using
an induction argument but I'm stuck... So every time one glues a cell then one needs
to show that this only throws in finitely m... | https://mathoverflow.net/users/11765 | On the fundamental group of a finite CW complex | Let $X$ be a CW-complex, and write $X\_k$ for the $k$-skeleton. The cellular approximation theorem says that any based map $S^1\to X$ is homotopic to a cellular map, and that any two cellular maps that are homotopic are homotopic via a cellular homotopy. This means that the map $\pi\_1(X\_1)\to\pi\_1(X\_2)$ is surjecti... | 19 | https://mathoverflow.net/users/10366 | 57591 | 35,893 |
https://mathoverflow.net/questions/57597 | 10 | Every countable order type, such as the countable ordinals, $\mathbb Z$, etc. can be embedded in $\mathbb Q$, so it is universal for countable order types. Is there a universal space for all linear orders of cardinality continuum? Or more generally, a universal space for all linear orders of any given cardinality?
| https://mathoverflow.net/users/4903 | Universal order type | You are looking for the concept of [saturated model](http://en.wikipedia.org/wiki/Saturated_model). A model $M$ is $\kappa$-saturated if any [type](http://en.wikipedia.org/wiki/Type_%28model_theory%29) consisting of fewer than $\kappa$ many assertions that is consistent with the elementary diagram of $M$ is realized in... | 24 | https://mathoverflow.net/users/1946 | 57599 | 35,899 |
https://mathoverflow.net/questions/57564 | 4 | Take the first $n$ primes $p\_1,...,p\_n$ and the primorial $P\_n$ .Denote by $p\_i$ every prime bigger than $p\_n$ and smaller than $P\_n$.
1) Is that true that there always be a number in any interval of consecutive integers of length $P\_n$ not divided by any $p\_i$? (It's the same as taking a residue class $r\_i\... | https://mathoverflow.net/users/14726 | residue classes of primes, covering intervals and bounds on the different ways | I did some computer programming to check plausibility. In future I request that you do this step yourself.
For $p\_n = 3$ and $P\_n = 6,$ the only prime in between is 5, and any interval of length 6 contains an integer not congruent to any prescribed value mod 5.
In C++ I was able to check up to 10,000,000. For de... | 3 | https://mathoverflow.net/users/3324 | 57603 | 35,902 |
https://mathoverflow.net/questions/57602 | 5 | Hello,
I've been reading the excellent online book on Algebraic Number Theory by J.S.Milne. In the section described above there is a footnote maintaining that the separability of the residue field extension implies the separability of the original field extension in the unramified case (base field complete w.r.t. a ... | https://mathoverflow.net/users/13462 | Unramified (finite) extensions of fields complete with respect to a discrete valuation | Let $R\to R'$ be a such extension of DVR, let $k'/k$ be the residue extension. Lift a generator of $k'/k$ to $\theta\in R'$. Then $R'=R[\theta]$ by Nakayama. The minimal polynomial $P(X)\in R[X]$ of $\theta$ has degree $[k':k]$, and it is separable because it is separable in the residue field (consider the GCD of $P(X)... | 9 | https://mathoverflow.net/users/3485 | 57606 | 35,905 |
https://mathoverflow.net/questions/57615 | 3 | The question is motivated by [this one.](https://mathoverflow.net/questions/57588/series-for-loglog-closed) It turned out (see my comment there) that the coefficients of the Taylor series for $\log\log x$ at $x=e$ have nice combinatorial description from [Sloane's encyclopedia](http://oeis.org/search?q=1%2C-2%2C7%2C-... | https://mathoverflow.net/users/nan | about power series for iterated logarithms | I think it may be simpler to deal with the Maclaurin series for the functions $\log(1-x)$, $-\log(1+\log(1-x))$, $-\log(1+\log(1+\log(1-x)))$, etc. The third one, for example, is the exponential generating function for $1,3,15,105,947,10472,137337,\dots$ which is <http://oeis.org/A000268> and there are a couple of refe... | 3 | https://mathoverflow.net/users/3684 | 57618 | 35,910 |
https://mathoverflow.net/questions/57377 | 8 | Following the following two previous questions on mathoverflow:
[Are all primes in a PAP-3?](https://mathoverflow.net/questions/34197/are-all-primes-in-a-pap-3/34298#34298)
and
[Covering the primes by 3-term APs ?](https://mathoverflow.net/questions/2214/covering-the-primes-by-3-term-aps)
I have attempted to sh... | https://mathoverflow.net/users/10898 | Are most primes in a prime arithmetic progression of length at least 3? | It is a [theorem of Ben Green](http://www.ams.org/mathscinet-getitem?mr=2180408) that every subset of the primes of positive relative density contains a progression of length three. As an immediate consequence, the set of primes $A$ which are not the first term in a progression of primes of length three has density zer... | 25 | https://mathoverflow.net/users/766 | 57635 | 35,920 |
https://mathoverflow.net/questions/57636 | 0 | Does anyone have a review of the article?
| https://mathoverflow.net/users/13466 | What is Deligne's article "Forms modulaires and representations de GL(2)" about? | Have you tried MathSciNet?
Here is the [review](http://www.ams.org/mathscinet-getitem?mr=347738), and hopefully the copyright daemons will not haunt me:
>
> This is a fairly detailed summary, with references, of the approach to modular or automorphic forms through the theory of group representations, in the case ... | 4 | https://mathoverflow.net/users/1409 | 57637 | 35,921 |
https://mathoverflow.net/questions/57630 | 10 | What is known about the order of $\zeta(1+it)$?
Iwaniec-Kowalski gives (pp. 226 citing a result of Vinogradov-Korobov)
$|\zeta(1+it)| \lesssim (\log t)^{2/3},$
and oppositely Titchmarsh gives (pp. 188 citing Bohr-Landau)
$|\zeta(1+it)| \gtrsim \log \log t$
for infinitely many values of $t$.
Is this the l... | https://mathoverflow.net/users/5621 | Order of $\zeta(1+it)$ | The theorem quoted from I-K is due to, and should therefore be attributed to, Vinogradov-Korobov. It is the best known unconditional result on the line $\mathrm{Re}(s)=1$, and is directly related to the width of the zero-free region by a result of Landau (see around 3.10 in Titchmarsh), so that any improvement of eithe... | 9 | https://mathoverflow.net/users/20038 | 57640 | 35,923 |
https://mathoverflow.net/questions/57653 | 16 | By a [semi-simplicial set](http://ncatlab.org/nlab/show/semi-simplicial+set) I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. Unlike when degeneracies are present, we have to include 1-dimensional ... | https://mathoverflow.net/users/49 | Degeneracies for semi-simplicial Kan complexes | The answer to (1) is to be found in
Rourke, C. P.; Sanderson, B. J.$\Delta$-sets. I. Homotopy theory.
Quart. J. Math. Oxford Ser. (2) 22 (1971), 321–338.
It is shown there that a Kan "semi-simplicial" set admits a compatible system of degeneracies.
By the way, the term "semi-simplicial" set is not the usual na... | 8 | https://mathoverflow.net/users/8032 | 57655 | 35,927 |
https://mathoverflow.net/questions/57643 | 9 | Let $w\ll n$ (say $w=n^{0.1}$) and $a\_1,\ldots,a\_w$ be positive real numbers such that $\sum\_{i \in w} a\_i=n$. Also, let $x\_1,x\_2,\ldots, x\_w$ be i.i.d. $\pm 1$ random variables. What is the best $t$ such that one always has
$$
\Pr\_{x\_1,\ldots, x\_w} [|\sum\_{i\le w} a\_i x\_i| \ge t] \ge 0.01
$$
| https://mathoverflow.net/users/10858 | Anti-concentration about the mean for sum of Bernoulli random variables | The best $t$ I do not know. But one can for example get some bounds by applying the Paley-Zygmund inequality to the random variable $Z=(\sum\_i a\_i X\_i)^2$.
| 2 | https://mathoverflow.net/users/12088 | 57658 | 35,928 |
https://mathoverflow.net/questions/57524 | 1 | I need the martingale part of the put payoff (not $C^2$..). Where $S\_t=exp(\sigma W\_t -\frac{\sigma^2t}{2})$
>
> $d[(S\_t -K)^+ ]$ ??
>
>
>
I guess I need to use local times but how?
| https://mathoverflow.net/users/13447 | Martingale part of the discontinuous put payoff | Thanks you all!
(proof, for $\phi(t,S\_t)=(K-S\_t)^+$:
**Step 1** *smoothing* :
$\phi\_\epsilon(x)=1\_{S\_t\leq K+\epsilon}\cdot\phi(x)+1\_{S\_t\in]K-\epsilon,K]} \cdot \psi(x)$, where $\psi(x)=-\frac{1}{\epsilon^2}(K-x)^2(K-x-2\epsilon)$ .
This function is $C^1$, and also $C^2$ excepting in a countable set.
*... | 2 | https://mathoverflow.net/users/13447 | 57659 | 35,929 |
https://mathoverflow.net/questions/57654 | 11 | The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a Banach space the same statement holds true.
Beyond the metrizable case this is no longer correct. So my first question is ... | https://mathoverflow.net/users/12482 | Example of noncomplete quotient of complete lcs mod closed subspace | A counterexample for both the first and third questions can be found in [*Counterexamples in Topological Vector Spaces*](http://rads.stackoverflow.com/amzn/click/354011565X) by Khaleelulla (p. 108).
Let $W$ denote the space of all $\mathbb C$-valued sequences $(x\_n)$ and $\Phi$ the space of finite sequences. Let $E=... | 4 | https://mathoverflow.net/users/5371 | 57660 | 35,930 |
https://mathoverflow.net/questions/57667 | 26 | Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?
| https://mathoverflow.net/users/2612 | Which Kahler Manifolds are also Einstein Manifolds? | This question can be interpreted in two different ways.
1) Which Kahler manifolds admit a Kahler metric that is at the same time Einstein?
2) Which Kahler manifolds admit an Einstein metric?
**If you want 1)**, then you need to start with a manifold whose canonical bundle is either a) ample (like hypersurfaces ... | 41 | https://mathoverflow.net/users/943 | 57669 | 35,936 |
https://mathoverflow.net/questions/17031 | 32 | This is maybe the first question I actually need to know the answer to!
Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of $\mathbb{H}/\Gamma(N)$ is generated by a single function. When $N = 2$, the cross-ratio $\lambda$ is such a function. A point of $\mathbb{H}... | https://mathoverflow.net/users/290 | Modular curves of genus zero and normal forms for elliptic curves | I think the answer to your question is the content of Velu's thesis: Courbes elliptiques munies d'un sous-groupe $Z/NZ\times \mu\_N$. In there, he explicitly writes down the universal elliptic curve over $X(p)$ for $p>3$.
| 11 | https://mathoverflow.net/users/92 | 57673 | 35,937 |
https://mathoverflow.net/questions/57559 | 5 | What is the classifying space for $S^1$-bundle? Here, $S^1$-bundle means a fiber bundle which doesn't mean that it is principal $S^1$-bundle.
I know that for a space $F$,
$\lbrace$the set of fiber bundles over $M$ whose fiber is $F\rbrace$ = $[M,B\operatorname{Homeo}(F)]$.
Therefore, my question can be rephrased... | https://mathoverflow.net/users/13453 | Classifying space for S1-bundle? | As mentioned by Tom Goodwillie, $Homeo(S^1)$ is homotopy equivalent to $O(2)=\mathbb Z/2\ltimes S^1$. We have a (split) short exact sequence of groups
$$
S^1 \to \mathbb Z/2 \ltimes S^1 \to \mathbb Z/2
$$
which, upon applying applying the functor $B$, produces a (split) fibration sequence of classifying spaces
$$
BS^1 ... | 8 | https://mathoverflow.net/users/5690 | 57676 | 35,939 |
https://mathoverflow.net/questions/57684 | 4 | Hi everybody,
I'm interested in the barycentric subdivision of a poset $P$, defined as the face poset of the corresponding order complex $\mathcal F(\Delta(P))$ (or alternatively, the poset of chains in $P$ ordered by inclusion). Actually, I'm interested in the subdivision of small categories, but I'm also happy unde... | https://mathoverflow.net/users/12879 | On the barycentric subdivision of a poset | This functor has no right adjoint, since it doesn't preserve colimits. Let $[m]$ denote the linearly ordered set $\{0,\ldots,m\}$ and consider the pushout of $[0]\overset{0}{\to}[1]$ and $[0]\overset{1}{\to}[1]$, which can be identified with $[2]$. Now $\mathrm{sd}[2]$ is the poset of chains in $[2]$, but the pushout o... | 4 | https://mathoverflow.net/users/12547 | 57694 | 35,951 |
https://mathoverflow.net/questions/57672 | 3 | My question is prompted by
[57589](https://mathoverflow.net/questions/57589/heuristic-behind-a-infty-algebras)
If $X$ is an object in a monoidal category with unit $I$ then $Y$ is a left dual
if we have $I\rightarrow Y\otimes X$ and $X\otimes Y\rightarrow I$ which satisfy
the well-known zig-zag identities.
My quest... | https://mathoverflow.net/users/3992 | What is a left dual up to homotopy? | The $\infty$-categorical version of the story is identical to the one you wrote (and explained in Lurie's Higher Algebra, Section 4.2.5): in any monoidal $\infty$-category there's a notion of left and right duals of objects, defined by data of evaluation and coevaluation maps together with zig-zag identities, and the l... | 3 | https://mathoverflow.net/users/582 | 57704 | 35,955 |
https://mathoverflow.net/questions/57692 | 8 | In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be the case, most often, that the interesting statements we try to force end up being forced by the whole poset.
A sufficien... | https://mathoverflow.net/users/7521 | Statements forced by one condition of a poset, but not the whole thing | There are several issues.
First, of course, if a statement $\varphi$ is forced by a condition $p\in\mathbb{P}$, then of course it is forced by every condition in the poset $\mathbb{P}|p$, which restricts $\mathbb{P}$ to conditions below $p$. So whenever a statement is forced by any condition, then there is a poset s... | 7 | https://mathoverflow.net/users/1946 | 57709 | 35,957 |
https://mathoverflow.net/questions/57711 | 0 | It is known
1. $P \subset P/poly$
2. $NP \not\subset P/poly \Rightarrow P \neq NP$
However, do we have a proof of:
$P \neq NP \Rightarrow NP \not\subset P/poly$ ?
I.e. is there a world where $P \neq NP$, but $NP \subset P/poly$?
Thanks!
| https://mathoverflow.net/users/3609 | "P vs NP" and "NP vs P/Poly" | No, it is unknown whether $P \neq NP \Rightarrow NP \not\subset P/Poly$. However, one may show that if $NP \subset P/Poly$ then the polynomial hierarchy collapses on the second level, what is rather unlikely.
| 10 | https://mathoverflow.net/users/13480 | 57716 | 35,961 |
https://mathoverflow.net/questions/57719 | 2 | Let $F$ be an arbitrary field of characteristic $0$, $K$ its algebraic closure. Define $M=\{ (x,y)\in M\_n(F)×M\_n(F) \mid [x,y]=0\}$ and let $N$ be the Zariski closure of $M$ in $K^{2n^2}$.
How can one show that $N$ contains the set $\{(axa^{-1},aya^{-1}) \mid (x,y)\in N, a\in \mathrm{GL}(n,K)\}$?
Thank you.
| https://mathoverflow.net/users/13481 | algebraic closure of commuting pairs of matrices | Note that $M$ is invariant under the $GL\_n(F)$-action given by $a \cdot (x,y):=(axa^{-1},aya^{-1})$. It follows that its closure $N$ is also invariant under $GL\_n(F)$. Since $F$ is infinite and $GL\_n$ is reductive, the rational points $GL\_n(F)$ are Zariski-dense in $GL\_n(K)$ by Borel, Linear Algebraic Groups, Corr... | 5 | https://mathoverflow.net/users/3380 | 57730 | 35,968 |
https://mathoverflow.net/questions/57723 | 11 | Let $G$ be a finite group. Is there necessarily a finite primitive permutation group $P$ and a normal subgroup $N>1$ of $P$ such that $P/N \cong G$?
If not, what restrictions are there on quotients of finite primitive permutation groups?
| https://mathoverflow.net/users/4053 | Is every finite group a proper quotient of a finite primitive group? | Yes. We can assume that $G$ is a transitive permutation group. Let $S$ be any primitive finite simple group, such as $A\_5$ in its natural representation. Now let $P$ be the wreath product of $S$ with $G$ using the product action, which has degree $d(P) = d(S)^{d(G)}$. This gives a primitive group, and the quotient of ... | 14 | https://mathoverflow.net/users/35840 | 57737 | 35,973 |
https://mathoverflow.net/questions/57725 | 10 | [Strassen Algoritm](http://en.wikipedia.org/wiki/Strassen_algorithm) is a well-known matrix multiplication divide and conquer algorithm.
The trick of the algorithm is reducing the number of multiplications to 7 instead of 8. I was wondering, can we reduce any further? Can we only do 6 multiplications?
Also, what happ... | https://mathoverflow.net/users/13482 | Strassen Algorithm 7 multiplications | You may be interested to know that there's a way to multiply $3\times3$ matrices using only 23 multiplications (where the naive method uses $27$). See Julian D. Laderman, A noncommutative algorithm for multiplying $3\times3$ matrices using $23$ muliplications, Bull. Amer. Math. Soc. 82 (1976) 126–128, MR0395320 (52 #16... | 16 | https://mathoverflow.net/users/3684 | 57740 | 35,975 |
https://mathoverflow.net/questions/57739 | 9 | Let $p$ be an odd prime number and consider the set of $p-2$ integers that is $\mathbb{Z}\_p$ minus 0 and 1. Next define two bijective functions on this set
\begin{align}
f(x) &= 1-x \mod p
\end{align}
and
\begin{align}
g(x) &= x^{-1} \mod p \qquad \text{(the multiplicative inverse of $x$).}
\end{align}
One can view th... | https://mathoverflow.net/users/9003 | Orbits in modular arithmetic | You can first notice that $fgfgfg(x)=x$ and conclude that most orbits have size $6$. It is easy to show that there is one orbit of size $3$ which you found and there is an orbit of size $2$ whenever there is a solution to $$x(1-x)\equiv 1\pmod{p}$$ this happens if $\binom{-3}{p}=1$ so if $p\equiv 1\pmod{3}$. You can ea... | 15 | https://mathoverflow.net/users/2384 | 57746 | 35,979 |
https://mathoverflow.net/questions/57749 | 6 | What do the eigenforms of the 1-form Laplace-de Rham operator look like on the 2-sphere, seen as vector fields via the inner product?
For the standard Laplace-de Rham operator on 0-forms (functions) the simple answer is the spherical harmonics. What about for the 1-form operator?
| https://mathoverflow.net/users/13491 | Laplace-deRham operator for 1-forms on the sphere | If $f:S^2\to R$ satisfies $\Delta f=\lambda f$, then
$$\Delta(df)=(dd^\*+d^\*d)(df)=\lambda df$$
and similarly
$$ \Delta(\ast df)=(dd^\*+d^\*d)(\*df)=\lambda \ast df $$
Since $H^1(S^2)=0$, these are all eigenvectors on 1-forms. Here $\*$ is the Hodge \* operator and $d^\*=-\ast d \ast$.
The vector field is the u... | 10 | https://mathoverflow.net/users/3874 | 57753 | 35,981 |
https://mathoverflow.net/questions/57741 | 30 | I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a simple way or point me to a source suitable for a non-expert mathematician? Thanks.
| https://mathoverflow.net/users/10076 | Algebraic P vs. NP | I suspect that the question under consideration is whether or not $VP=VNP$; this is the problem directly studied by geometric complexity theorists, as I understand their work. This project is described in some detail [here](https://doi.org/10.1137/090765328 "An overview of mathematical issues arising in the geometric c... | 28 | https://mathoverflow.net/users/6950 | 57758 | 35,984 |
https://mathoverflow.net/questions/57766 | 38 | On math.stackexchange [it was asked](https://math.stackexchange.com/questions/17062/is-there-essentially-only-1-jordan-arc-in-the-plane/17080#17080) whether all arcs in the plane are ambient-isotopic. I suggested that one could prove this by appealing to the Schönflies theorem, which you can do as long as you can exten... | https://mathoverflow.net/users/9417 | Why are there no wild arcs in the plane? | One way to show all arcs are tame is to apply the Riemann mapping theorem to the complement of the arc on $S^2$. Caratheodory proved that whenever the complement of a simply-connected domain is locally connected, then the Riemann mapping extends to a continuous map of the disk to the plane. One half of the disk paramet... | 31 | https://mathoverflow.net/users/9062 | 57770 | 35,989 |
https://mathoverflow.net/questions/57769 | 12 | This is kind of embarrassing, but I never figured out how to cite journal names in the bibliography, especially when to abbreviate and how. For example, do we write "Adv. in Math." or "Advances in Math."? Or is "Trans. of AMS" OK? So:
>
> Is there any formal guideline on how to cite and abbreviate journals names in... | https://mathoverflow.net/users/2083 | How to cite math journals? | Hailong, I would still suggest following MathSciNet, but you don't necessarily have to go there every time. Presumably the journals you cite come from a small finite set. This may be a pain at first, but you will get them all very soon.
Actually, I use BibTeX and so I have a growing database of papers. You can go to ... | 15 | https://mathoverflow.net/users/10076 | 57772 | 35,991 |
https://mathoverflow.net/questions/57752 | 2 | Hello all,
I am interested in the following question. Suppose a,b,c,z are points in the complex sphere. Consider the family of curves g through a,b,c, and for each g let U be the complement of g in the sphere. Can we find g so that p\_U(z) is minimal for this family of curves, where p\_U represents the hyperbolic(or ... | https://mathoverflow.net/users/13358 | Question concerning minimum of hyperbolic metric | If I understand your question correctly, the "complex sphere" is the same as the Riemann sphere, or the one point compactification of $\mathbb C$.
There is a well-developed theory of questions like this. Some of the keywords are Jenkins-Strebel differential, measured foliations, extremal length, fatgraphs, Schwarz-Ch... | 2 | https://mathoverflow.net/users/9062 | 57775 | 35,993 |
https://mathoverflow.net/questions/57657 | 32 | A short version of my question is: Is there a $p$-adic theory of integration?
Now let me expand a little further. In introductory texts such as Koblitz' book $p$-adic numbers,.. a bunch of $p$-adic analysis is developed. However, since all applications are towards number theory, the exposition stops at some point. In... | https://mathoverflow.net/users/3757 | $p$-adic integrals and Cauchy's theorem | There is an important difference, relevant to the original question, between the two kinds of $p$-adic integrals mentioned by Kevin in his comments. Because I see frequent confusion on this issue, I thought I'd comment.
The 'usual' $p$-adic integrals as you might see in, say, Tate's thesis on L-functions or the adeli... | 41 | https://mathoverflow.net/users/1826 | 57781 | 35,995 |
https://mathoverflow.net/questions/57787 | 2 | In general when one proves that the product of semisimple (i.e. diagonalizable) matrices is semisimple one assumes they commute and are thus simultaneously diagonalizable, and then the result follows. I was wondering if anyone knew of an example of non-commuting semisimple matrices whose product is not semisimple.
| https://mathoverflow.net/users/13139 | The product of non-commuting semisimple matrices need not be semisimple | How about $\begin{pmatrix} 5 & 3 \\\ 8 & 5 \end{pmatrix}$ and
$\begin{pmatrix} 2 & -3 \\\ -3 & 5 \end{pmatrix}$?
| 5 | https://mathoverflow.net/users/3992 | 57792 | 36,002 |
https://mathoverflow.net/questions/57794 | 1 | Hello,
I'm trying to solve the following integral :
$\int\_0^\infty \frac{1}{t^{d/2}}(e^{-\gamma t} - e^{-\delta t})dt$.
I know it equals
$\Gamma(1-\frac{d}{2})[\gamma^{\frac{d}{2}-1}-\delta^{\frac{d}{2}-1}]$ for every $d<4$.
However, this does not work for $d=2$ as the gamma function is not defined in zero.... | https://mathoverflow.net/users/13472 | Undefined gamma function problem | A direct calculation for $d=2$ is also possible and interesting:
Let $G(\lambda, \mu) = \int\_0^\infty \frac{e^{- \lambda t} - e^{- \mu t}}{t} dt$, for $\lambda, \mu > 0$. There are no problems at $t=0$ because $e^{- \lambda t} - e^{- \mu t} = (\mu - \lambda)t + O(t^2)$ near zero.
Clearly $G(\lambda, \mu) = G(\lamb... | 5 | https://mathoverflow.net/users/6651 | 57800 | 36,009 |
https://mathoverflow.net/questions/57761 | 7 | Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL\_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequently, the character group $X(G)$ of all algebraic group homomorphisms $G \rightarrow \mathbb{G}\_m$ is trivial, because any character $\ch... | https://mathoverflow.net/users/7932 | Character group of Frobenius kernels | It's probably most natural to consider this as a question about the (rational) representations of Frobenius kernels, in the spirit of Jantzen's book *Representations of Algebraic Groups* (Chapter II.3). Given a connected, simply connected semisimple group $G$, the irreducible representations of its Frobenius kernel $G\... | 4 | https://mathoverflow.net/users/4231 | 57815 | 36,014 |
https://mathoverflow.net/questions/57806 | 4 | Does there are exist simple proof for the following statement?
Let $\rho,V$ be an irreducible representation of group $G$ of dimention $n$.
Assume that there are exist $g \in G$ such that $\rho(g)$ just flips two coordinates. (that is
$\rho(g)e\_1=e\_2,\ \rho(g)e\_2=e\_1,\ \rho(g)e\_i=e\_i$) Then $|G|\geq 2^n$, where... | https://mathoverflow.net/users/4246 | Irreducible representation flipping two elements | [I assume characteristic 0, and then I can extend the scalars to an algebraically closed field, which I can take to be $\mathbb{C}$ as $G$ is finite and $V$ is finite dimensional. It may appear that I then need the starting $V$ to be *absolutely* irreducible for the argument below, but actually since the eigenvector $e... | 11 | https://mathoverflow.net/users/11108 | 57824 | 36,020 |
https://mathoverflow.net/questions/57819 | 5 | what is the best available approximation ( say up to 10 digits ) for LambertW(x) or exp(LambertW(x)) for x > 2000
| https://mathoverflow.net/users/13452 | best approximation to the LambertW(x) or exp(LambertW(x)) | **Extended answer**
The approximation described below is original, explicit (in some sense), and very accurate.
It is closely related to [this question](https://mathoverflow.net/questions/45390/another-special-property-of-the-exponential-function) (second paragraph).
So, you want to approximate the solution $W(x)$... | 8 | https://mathoverflow.net/users/10227 | 57861 | 36,042 |
https://mathoverflow.net/questions/57825 | 13 | Let $N\_d$ denote the number of rational curves in $\mathbf P^2$ passing through $3d-1$ points in general position. Maxim Kontsevich discovered a famous recursion for these numbers:
$$ N\_d = \sum\_{k+l = d} N\_k N\_l k^2 l \left( l \binom{3d-4}{3k-2} - k \binom{3d-4}{3k-1}\right).$$
The proof of this recursion goes b... | https://mathoverflow.net/users/1310 | Who streamlined Kontsevich's count of rational curves? | In Manin's book "Frobenius manifolds, quantum cohomology, and moduli spaces", section 0.6.2, the argument ("an old trick of enumerative geometry") is attributed to Kontsevich. I would guess that Kontsevich probably came up with the "streamlined proof" but preferred to talk about it within the framework of WDVV because ... | 12 | https://mathoverflow.net/users/83 | 57864 | 36,045 |
https://mathoverflow.net/questions/57802 | 4 | This question is somehow related to the question [What properties define open loci in excellent schemes?](https://mathoverflow.net/questions/16235/what-properties-define-open-loci-in-excellent-schemes%20%22What%20properties%20define%20open%20loci%20in%20excellent%20schemes?%22).
Let $f:X\to S$ be a proper (or even pr... | https://mathoverflow.net/users/3847 | What properties define open loci in families? | The recentish book of Görtz and Wedhorn (see <http://www.algebraic-geometry.de/> ) has an Appendix E which gives a long list of properties of morphisms for which the corresponding set of the base is open or constructible (when only constructibility holds), together with references for the proofs (either to their book o... | 4 | https://mathoverflow.net/users/20038 | 57867 | 36,047 |
https://mathoverflow.net/questions/57808 | 17 | Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ?
I first asked this question at math.stackexchange.com and was suggested to post it here : <https://math.stackexchange.com/questions/25519/correlated... | https://mathoverflow.net/users/12383 | Correlated Brownian motion and Poisson process | You can not construct a Poisson process $N$, and a Brownian motion $W$ on the same filtration such that they are dependent. They are always independent on the given filtration.
Let $(W\_t)\_{t\geq 0}$ be a standard Brownian motion and $(N\_t)\_{t\geq 0}$ be a Poisson process with intensity $\lambda$, both defined on... | 17 | https://mathoverflow.net/users/13486 | 57868 | 36,048 |
https://mathoverflow.net/questions/57820 | 41 | Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements that these structures can possibly make. Global space is construed as patches connected via transport, which identifies me... | https://mathoverflow.net/users/nan | Is there a mathematical axiomatization of time (other than, perhaps, entropy)? | In probability, time is usually handled as a nested sequence of $\sigma$-algebras (say $B\_t$, with $B\_t \subset B\_s$ if $t\leq s$), and to find the reality (call the reality $f$, and it includes the state at all times past and future) at time $t$, one takes the conditional expectation $f\_t := E[f | B\_t ]$. The seq... | 29 | https://mathoverflow.net/users/935 | 57869 | 36,049 |
https://mathoverflow.net/questions/57870 | 2 | Is there a general way of counting the number of homogeneous polynomials of certain degree in a complex projective space or a weighted complex projective space, mod the ideal generated by some homogeneous polynomials with smaller degrees?
As an example, consider the degree 18 homogeneous polynomials in $W\mathbb{P}\... | https://mathoverflow.net/users/7035 | Count the number of homogeneous polynomials | I guess what you are after is the Hilbert function of ideal. More precisely, if you have a multigraded polynomial ring $k[x\_1,\ldots,x\_n]$ and the ideal is given by $I=(f\_1,\ldots,f\_r)$, the Hilbert function is simply the number
$$
h(\alpha)=\dim\_k (k[x\_1,\ldots,x\_n]/I)\_\alpha.
$$Here $\alpha$ may take values i... | 8 | https://mathoverflow.net/users/3996 | 57874 | 36,052 |
https://mathoverflow.net/questions/57811 | 5 | Hello,
Here is a problem I encountered in the study of Kähler manifolds but there is a natural generalisation of this for topological spaces.
If $X$ is a topological space, denote by $g\_\mathbb{R}$ the real genus of $X$, that is the maximal dimension of an isotropic subspace in $H^1(X,\mathbb{R})$ (isotropic means t... | https://mathoverflow.net/users/12517 | Isotropic subspaces in cohomology | In the full linear algebra generality, the answer is no.
Take four generic $2$-planes, $V\_1$, $V\_2$, $V\_3$ and $V\_4$ in $\mathbb{R}^4$. Over $\mathbb{C}$, there are always two $2$-planes $W$ such that $W \cap V\_i \neq (0)$ for $1 \leq i \leq 4$. (This is the first nontrivial Schubert calculus example.) Choose t... | 3 | https://mathoverflow.net/users/297 | 57878 | 36,054 |
https://mathoverflow.net/questions/57880 | 4 | Does there exist a function f s.t.:
(1) $f(f(n)) \in O(f(n))$
(2) $f(n) \in \Omega(\cup\_i n^i)$
Thanks!
| https://mathoverflow.net/users/3609 | Growth of Functions | The answer is no. By (1), we know that $f(f(n))\leq k f(n)$ for some constant $k$ and sufficiently large $n$. By (2), we get in particular that $c n^2\leq f(n)$ for some constant $c$ and sufficiently large $n$. This implies that $f(n)$ eventually gets large, and so applying it again we get $c (f(n))^2\leq f(f(n))$ for ... | 12 | https://mathoverflow.net/users/1946 | 57882 | 36,057 |
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