parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/217935 | 9 | [Parse it as *(locally compact)ly* generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here [before](https://mathoverflow.net/questions/47702/why-the-w-in-cgwh-compactly-generated-weakly-hausdorff-spaces). As an a... | https://mathoverflow.net/users/78849 | Is every locally compactly generated space compactly generated? | The paper "[A distinguishing example in k-spaces](http://www.ams.org/journals/proc/1987-100-03/S0002-9939-1987-0891170-3/)" by John Isbell constructs an example of a locally compact space $X$ which is not compact-Hausdorffly generated.
| 8 | https://mathoverflow.net/users/51164 | 217940 | 102,387 |
https://mathoverflow.net/questions/217956 | 11 | I have received [a complaint](https://mathoverflow.net/questions/80865/least-prime-in-a-arithmetic-progression/80867#comment537038_80867) about my 2011 answer
[least prime in a arithmetic progression](https://mathoverflow.net/questions/80865/least-prime-in-a-arithmetic-progression/80867#80867)
which, indeed, give... | https://mathoverflow.net/users/3324 | Update for 2015: least prime of form nq+1, with q prime? | The most optimistic conjecture is that the least prime in this (or indeed any progression $a\pmod q$) is $\ll q (\log q)^2$. This is an analog of Cramer's conjecture on primes in short intervals, so way beyond reasonable conjectures like GRH!
On GRH [Lamzouri, Li, and Soundararajan](http://arxiv.org/pdf/1309.3595v1.... | 17 | https://mathoverflow.net/users/38624 | 217961 | 102,394 |
https://mathoverflow.net/questions/217955 | 1 | Let $(X,\preceq)$ be a poset.
>
>
> >
> > Is there a standard, generally recognised term for a set $A \subset X$ satisfying
> > $$ \forall x,y,z \in X, \ (x \in A \ \textrm{ and } \ z \in A \ \textrm{ and } \ x \preceq y \preceq z) \ \Rightarrow \ y \in A \ ?$$
> >
> >
> >
>
>
>
Of course, the natural te... | https://mathoverflow.net/users/15570 | "Interval" terminology for (partially) ordered sets | These sets are usually called convex sets. See for example [this paper](http://dml.cz/bitstream/handle/10338.dmlcz/118617/CommentatMathUnivCarolRetro_34-1993-3_23.pdf).
| 3 | https://mathoverflow.net/users/22277 | 217967 | 102,395 |
https://mathoverflow.net/questions/217881 | 4 | My question is about the passage (11.1) in the book of Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$, an algebraic stack $\mathscr{X}$ over $S$, and a point $\xi$ of $\mathscr{X}$. The residual gerbe $\mathscr{G}\_\xi$ at $\xi$ is defined as follows: choose an $S$-field $K$ and an $S$-morphism... | https://mathoverflow.net/users/70964 | Is the residual gerbe really independent of the choice of a representative? | IIRC a reference for a discussion is David Rydh's paper \'Etale d\'evissage, descent and pushouts of stacks, Appendix B. (Of course grghxy also already answered your question too, my apologies to grghxy.)
You can look at [Tag 06ML](http://stacks.math.columbia.edu/tag/06ML) for an answer to your question if you alread... | 3 | https://mathoverflow.net/users/60618 | 217975 | 102,399 |
https://mathoverflow.net/questions/217829 | 4 | For every prime $p$, does there exists integers $x\_1$, $x\_2$ and $x\_3$ ($0\leq x\_1, x\_2, x\_3 \leq \lfloor cp^{1/3}\rfloor$ and $c$ is some large constant) such that $\frac{p-1}{2}-\lfloor 2cp^{1/3} \rfloor \leq f(x\_1,x\_2,x\_3) \leq \frac{p-1}{2}$, where, $f(x\_1,x\_2,x\_3)=x\_1+x\_2+x\_3+2(x\_1x\_2+x\_2x\_3+x\_... | https://mathoverflow.net/users/70652 | Integer solution | I doubt that the lower bound $\frac{p-1}{2} - 2cp^{1/3}$ holds for all $p$. Here is a proof for the weaker bound $\frac{p-1}{2} - cp^{1/2}$.
First of all, the inequality $\frac{p-1}{2}-cp^{1/2} \leq f(x\_1,x\_2,x\_3) \leq \frac{p-1}{2}$ is essentially equivalent to
$$p- O(p^{1/2}) \leq (2x\_1+1)(2x\_2+1)(2x\_3+1) \l... | 0 | https://mathoverflow.net/users/7076 | 217983 | 102,402 |
https://mathoverflow.net/questions/217982 | 1 | Let $f: M \to N$ be a diffeomorphism between two riemannian Manifolds. Suppose there exist constants $0 < c \leq C$ such that for all $p \in M$, we have $c \leq |df\_p| \leq C$. Here, $df$ denotes the differential of $f$ as usual and $|df\_p|$ denotes the energy density of $f$ at $p$, i.e, the norm coming from the indu... | https://mathoverflow.net/users/78554 | Estimate for differential of inverse map | Lower bound yes. Upper bound no.
If you take orthonormal frames for $M$ and $N$, then the norm $|df\_p|$ is the Frobenius norm on the matrix expression in those frames. Note that in any bases $df\_p$ and $df^{-1}\_{f(p)}$ are inverse matrices. The upper bound on the Frobenius norm of $df\_p$ implies that each of the... | 1 | https://mathoverflow.net/users/3948 | 217985 | 102,403 |
https://mathoverflow.net/questions/217805 | 4 | Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about this construction:
1. When is it the case that $X/Y$ is again an $n$-fold loop space?
2. Are there weaker conditions th... | https://mathoverflow.net/users/11546 | When is the quotient by an $n$-fold loop space an $m$-fold loop space? | I think what Jesper and Amrani wrote should clarify why taking the cofiber is not very appropriate in this context and what Qiaochu wrote clarifies why taking the fiber of the delooping is "better".
Let me try to complement the picture. I apologize in advance for the long answer, but since we already talked about it... | 6 | https://mathoverflow.net/users/33199 | 217988 | 102,405 |
https://mathoverflow.net/questions/217977 | 8 | Suppose $(C,\odot,\Bbb I)$ is an additive category with a compatible symmetric monoidal structure and $Pic(C)$ is the group of isomorphism classes of objects which have an inverse under $\odot$. For $\alpha \in Pic(C)$, we would like define the $\alpha$'th coefficient functor by choosing a representative $P^\alpha \in ... | https://mathoverflow.net/users/360 | Obstructions to Picard-graded groups of maps | Let $\mathbf{Pic}(C)$ be the monoidal subgroupoid of tensor invertible objects and isomorphisms between them, so $\pi\_0\mathbf{Pic}(C)=Pic(C)$. For the definition of those appropriate maps, it is clearly necessary and sufficient to choose representatives in such a way that we get a monodical splitting $Pic(C)\to \math... | 3 | https://mathoverflow.net/users/12166 | 217992 | 102,406 |
https://mathoverflow.net/questions/217927 | 49 | 1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety its algebra of regular functions to $\mathbb{A}^1$, and to each finitely generated reduced algebra its (ringed) space of ... | https://mathoverflow.net/users/51164 | Why is there a duality between spaces and commutative algebras? | I don't claim to have a complete answer but here are some miscellaneous comments.
1. Note that topological spaces are already very nearly defined to be dual to certain commutative algebra-like structures, namely their [frames](http://ncatlab.org/nlab/show/frame) of open subsets. The simplest interesting case of this... | 18 | https://mathoverflow.net/users/290 | 217998 | 102,409 |
https://mathoverflow.net/questions/218000 | 1 | I cannot find a proof of this theorem. May anyone assist?
>
> $p\_{n+1}-p\_n\gg\frac{\log \log \log p\_n}{\log \log \log \log p\_n} \log{p\_n}$
>
>
>
| https://mathoverflow.net/users/75039 | Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$ | The bound as stated is false, because not all prime gaps are that large. In fact we know since the work of Yitang Zhang (2013) that there are infinitely many bounded prime gaps.
The state of the art regarding (occasional) large prime gaps is contained in the work of [Ford-Green-Konyagin-Maynard-Tao](http://de.arxiv.o... | 9 | https://mathoverflow.net/users/11919 | 218001 | 102,411 |
https://mathoverflow.net/questions/217919 | 7 | According to [PolyMath](http://michaelnielsen.org/polymath1/index.php?title=Finding_primes)
>
> **(Strong) conjecture**. There exists *deterministic* algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in k. You may assume as many standard conjec... | https://mathoverflow.net/users/12481 | What is wrong with this deterministic algorithm efficiently generating large primes? | I have a somewhat different answer. The bound (1) is not known, but it is expected by some optimist paper. See for example this [paper](http://www.ams.org/journals/mcom/1996-65-216/S0025-5718-96-00763-6/S0025-5718-96-00763-6.pdf) which says that one expects $p' = O(p (\log p)^2)$ and gives (page 1718) three references ... | 11 | https://mathoverflow.net/users/9317 | 218009 | 102,414 |
https://mathoverflow.net/questions/215809 | 5 | Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a\_1,a\_2,\dots a\_n \in G$ such that
$$x^{a\_1}x^{a\_2}\dots x^{a\_n}=1,$$
for all $x\in G$.
1. Assume that $G$ has order $pq$, where $p\neq q$ are primes. Can $G$ satisfy a generalized identity of degree $p$?
I... | https://mathoverflow.net/users/31883 | Generalized identities of (soluble) groups | Assume $G=\langle a,b:a^p=b^q=1,a^b=a^\lambda\rangle$ is a non-abelian group of order $pq$ ($p>q$). Suppose $G$ satisfies the following generalized identity of degree $k$,
\begin{equation}
x^{g\_1}\cdots x^{g\_k}=1.\quad(\*)
\end{equation}
Taking modulo $G'$, one observe that $k$ is a multiple of $q$. Now, assume $g\_i... | 1 | https://mathoverflow.net/users/40723 | 218014 | 102,417 |
https://mathoverflow.net/questions/218013 | 1 | Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question:
For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?
| https://mathoverflow.net/users/36688 | Retract embedding of $S^{n}$ in its unit tangent bundle | Here is a homological argument. The unit tangent bundle is a spherical fibration
$$
S^{n-1} \to E \stackrel{p}{\to} S^n
$$
and so it has a long exact Gysin sequence in homology, part of which looks like
$$
\cdots \to H\_1(S^n) \to H\_n(E)\stackrel{p\_\ast}{\to} H\_n(S^n) \stackrel{\cap e}{\to} H\_0(S^n)\to \cdots .
$$... | 3 | https://mathoverflow.net/users/8103 | 218018 | 102,420 |
https://mathoverflow.net/questions/218007 | 6 | Suppose $X$ is an algebraic curve, $F$ its function field, $\mathbb{A}\_F$ its adele, $O\_x$ the ring of integers at local field of $x$. Why does $GL\_n(F)\backslash GL\_n(\mathbb{A}\_F)/\prod\_xGL\_n(O\_x)$ classify vector bundles over $X$?
| https://mathoverflow.net/users/nan | Why does $GL_n(F)\backslash GL_n(\mathbb{A}_F)/\prod_xGL_n(O_x)$ classify vector bundles over $X$? | This is a good exercise. Here's a big hint for one direction. Given a rank $n$ vector bundle, write down an isomorphism between it and the trivial rank $n$ vector bundle on the generic point of the curve (note: different choices will differ by an element of $GL\_n(F)$). Now write down an isomorphism between it and triv... | 4 | https://mathoverflow.net/users/43076 | 218029 | 102,424 |
https://mathoverflow.net/questions/208406 | 31 | Persistent homology is a well-developed tool which allows topological analysis of large data sets. From a topological perspective, the input is a filtered complex, and the output is a sequence of collections of intervals (one for each dimension) called a *persistence barcode*. The barcode gives information about homolo... | https://mathoverflow.net/users/8103 | Persistence barcodes and spectral sequences | The answer to your question is no, nobody has used persistence to improve the algorithmic efficiency of computing differentials, although of course the relationship between persistence intervals of a filtration and various terms in its Leray spectral sequence have been [described rather explicitly](http://arxiv.org/abs... | 18 | https://mathoverflow.net/users/18263 | 218030 | 102,425 |
https://mathoverflow.net/questions/217950 | 4 | Let $\mathfrak{gl}\_n(\mathbb{R})$ be the Lie algebra of matrices with real entries and $GL\_n(\mathbb{R})$ its associated Lie group. Recall that a linear subgroup $G \subseteq GL\_n(\mathbb{R})$ acts by conjugation on $\mathfrak{gl}\_n(\mathbb{R})$, that is, for $g \in G$ its action on $A \in \mathfrak{gl}\_n(\mathfra... | https://mathoverflow.net/users/70498 | Invariant polynomials with respect to group actions on matrices | For the action of the upper unipotent group, which we shall denote by $G$ here, one can continue in the following way: Let $g\_r = \pmatrix{1 & r \\ 0 & 1}\in G$. Consider a $2\times 2$ real matrix $M=\pmatrix{a & b \\ c & d }$. Then $g\_rMg\_r^{-1} = \pmatrix{a+rc & b-r(a-d) - r^2c \\ c & d-rc}$. Let us write $x=a+d$ ... | 2 | https://mathoverflow.net/users/41644 | 218047 | 102,432 |
https://mathoverflow.net/questions/201587 | 32 | The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the circle with its Lie group framing. It is well known that this element $\eta$ in stable homotopy (like all elements of posi... | https://mathoverflow.net/users/22 | Nilpotence of the stable Hopf map via framed cobordism | Answer Summary
--------------
Let $\eta$ be the framed 1-manifold which is the Lie group framing on the circle and let $\nu$ be the Lie group framing on $S^3 = Spin(3)$. I am probably going to conflate these framed manifolds with their classes in frame cobordism. I hope you forgive me.
There are lots of geometric ... | 26 | https://mathoverflow.net/users/184 | 218053 | 102,434 |
https://mathoverflow.net/questions/214812 | 8 | Does there exist the set of balls(may be not disjoint) $X=\{B\_i\subset\mathbb{R^2};i\in I\}$, satisfing following properties?(Note that the ball has a positive real radius)
1. Let the set of all lines in plane to be $L$. For each $l \in L,\ l\cap B\_i \ne \varnothing $ for some $i\in I$.
2. $\{N\_l=\text{the number... | https://mathoverflow.net/users/74273 | Set of balls which the number of the ball intersects lines on the plane is bounded | No, such $X$ does not exist.
Assume the contrary. Take a point $O$ which is outside all disks and a circle $\omega$ centered at $O$. We implement polarity with respect to this circle (thus, in what follows we consider only lines not passing through $O$).
Take any ball $B\_i\in X$. The poles of all lines tangent to ... | 2 | https://mathoverflow.net/users/17581 | 218063 | 102,438 |
https://mathoverflow.net/questions/218002 | 13 | There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct Eilenberg-MacLane spectra $HA$, where $A$ is an abelian group. In $S$-modules, this is described in Section IV.2 of this vers... | https://mathoverflow.net/users/16109 | Fibrant-cofibrant models of Eilenberg-MacLane spectra | The following four categories are models for spectra with Eilenberg-MacLane spectra of the desired form.
1. Kan's category of semisimplicial spectra [1]
2. The category $\mathbf{Sp}^\mathbb{N}(\mathbf{\Sigma})$ of sequential spectra of pointed simplicial sets together with the Kan suspension $\Sigma$
3. The category ... | 12 | https://mathoverflow.net/users/80168 | 218069 | 102,441 |
https://mathoverflow.net/questions/217923 | 8 | I am reading Witten's [paper](https://projecteuclid.org/euclid.cmp/1104161738) on topological field theories, in specific the topological twist in page 359. In order to perform the twist he takes the diagonal subgroup of $K = SU(2)\_{\text{Right}} \times SU(2)\_{\text{Isospin}}$ where the first comes from the $SU(2)\_{... | https://mathoverflow.net/users/80109 | What does it mean to take the diagonal of the group $SU(2) \times SU(2) $? | As noted in Vit's post, the diagonal group $G\_D \subset G \times G$ consists of the elements $\{(g,g) : g \in G \}$ with multiplication defined to be $(g,g) \cdot (h,h) = (gh,gh)$. A representation $(R\_1,R\_2)$ of $G \times G$ thus transforms as the representation $R\_1 \otimes R\_2$ under $G\_D$, where $\otimes$ den... | 9 | https://mathoverflow.net/users/13731 | 218080 | 102,444 |
https://mathoverflow.net/questions/214654 | 8 | My question can be viewed as a generalization of the four-color problem. Instead of a planar graph, consider a triangulation of a d-sphere. One wants to color vertices with N colors so that no two vertices connected by a 1-simplex have the same color. For a fixed d, is there an N which suffices for any triangulation? W... | https://mathoverflow.net/users/26536 | Colorings of triangulations as a generalization of the four-color problem | When $d>2$, $N=\infty$. I got this fact from [this preprint of Lutz and Möller](http://arxiv.org/abs/1503.08251) which discusses "higher" coloring problems in manifolds.
There are at least two routes to this:
First, as mentioned as (3) after Definition 1.1, [Walkup proved](http://link.springer.com/article/10.1007%... | 2 | https://mathoverflow.net/users/353 | 218081 | 102,445 |
https://mathoverflow.net/questions/218071 | 13 | In a [previous Mathoverflow question](https://mathoverflow.net/questions/217730/classifiying-sphere-eversions), we saw that the fundamental group of the space $Imm(S^2,\mathbb{R}^3)$ of immersions the 2-sphere in ordinary 3-space is isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}$.
We also saw that rotating the sphere ... | https://mathoverflow.net/users/58307 | Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphisms of the first | The action of $\text{Diff}^+(S^2)$ on $\text{Imm}(S^2,\Bbb R^3)$ is free. For pick any immersion $i$ and diffeomorphism $f$; given $x \in \Bbb R^3$ $i^{-1}(x)$ is a closed discrete (because $i$ is an immersion) set and hence finite. So if $if = i$, $f$ has finite orbits, and by [this previous MathOverflow question](htt... | 10 | https://mathoverflow.net/users/40804 | 218089 | 102,449 |
https://mathoverflow.net/questions/218076 | 6 | Is an infinite topological direct sum of amenable Banach algebras amenable again?
Can you give me a good reference about this notion?
Thanks
| https://mathoverflow.net/users/27066 | Infinite topological direct sum of amenable Banach algebra | The OP has now clarified that he or she is asking about $c\_0$-direct sums. To answer the question we need the notion of the amenability constant of a Banach algebra. Everything that follows is probably "folklore", in the sense that
(i) Johnson knew how to do it in 1972 but, due to the different culture at the time o... | 8 | https://mathoverflow.net/users/763 | 218096 | 102,452 |
https://mathoverflow.net/questions/210287 | 3 | For a nonlinear optimization problem having only linear constraints, by the Linearity Constraint Qualification, no further constraint qualification is required for the Karush-Kuhn-Tucker (KKT) conditions to hold. <https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions> . In particular, the Jacobian... | https://mathoverflow.net/users/75420 | Valid KKT Constraint Qualification? Linear constraints not full rank, Jacobian of nonlinear constraints full rank and independ. of linear constraints | This is true. It's a particular case of the Constant Rank Constraint Qualification from Janin.
| 3 | https://mathoverflow.net/users/1172 | 218097 | 102,453 |
https://mathoverflow.net/questions/218072 | 3 | It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses.
Also it is possible to construct straight line containing given point and parallel to given straight line by using only a ruler, with the condition that we can measure out ... | https://mathoverflow.net/users/73577 | Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses? | Yes, this is possible.
First, note that it is possible to construct a pair of perpendicular lines. To do this, make a rhombus. (On two lines that meet at a point $A$, construct segments of the same length starting from $A$. Then constructing parallels yields a parallelogram with two adjacent sides of equal length. Th... | 4 | https://mathoverflow.net/users/68305 | 218106 | 102,455 |
https://mathoverflow.net/questions/218100 | -2 | Let
$\theta(x)=\begin{cases}
0 & \text{ if } x<0 \\
1 & \text{ if } x\ge 0
\end{cases}$
Do you know any way to calculate this number:
$$\sum\_{r=493701}^{506199}\sum\_{k=0}^{100}(-1)^k\frac{\binom{100}{k}\binom{r-10001k+99}{99}\theta(r-10001k+99)}{\binom{r+99}{r}}$$
or estimate it within 0.01. Gap and wolfram f... | https://mathoverflow.net/users/47958 | Calculating a sum including large numbers | By comments after adding mpmath package to python one can run this program to see the result:
```
from mpmath import *
mp.dps=50
def th(x):
if x<0:
return 0
else:
return 1
def f(r,k):
a=(-1)**k*binomial(100,k)*binomial(r-10001*k+99,99)*th(r-10001*k+99)
a= a/binomial(r+99,r)
ret... | 2 | https://mathoverflow.net/users/47958 | 218123 | 102,463 |
https://mathoverflow.net/questions/218120 | 9 | Let $H:=\mathbb{Z}\*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I want to know if $H$ is a ($\mathbb{Z}$)linear group that is to say is there an injective homomorphism $f: H\to GL\_m(\mathbb{Z})$ for $m\geq n.$
I asked the question on Math Stack Exchange (<https://math.stackexchange.com/questions/1430677/is-the-... | https://mathoverflow.net/users/73395 | Is the free product $\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}$ linear over $\mathbb{Z}$? | The group $Z\*Z/n$ is virtually free: the kernel $K$ of the projection to $Z/n$ is free of index $n$ (say by the Kurosh theorem and since each finite order element is conjugate to an element of $Z/n$). Since a free group has a faithful representation over $Z$ of degree 2, the induced representation of this representati... | 14 | https://mathoverflow.net/users/15934 | 218129 | 102,465 |
https://mathoverflow.net/questions/218092 | 10 | Is there a good method to estimate the diameter of a closed hyperbolic 3-manifold?
I am particularly interested in know the diameter of the Weeks manifold.
| https://mathoverflow.net/users/50629 | Diameter of hyperbolic 3-manifolds | On page 356 of his paper *[The ortho-length spectrum for hyperbolic 3-manifolds](http://qjmath.oxfordjournals.org/content/47/3/349.extract)*, Meyerhoff shows that if $M$ is a closed hyperbolic $3$-manifold of volume $V$ then its diameter $\mathrm{diam}(M)$ satisfies $$\mathrm{diam}(M) < \frac{V}{\pi\sinh^2(\ell/4)},$$
... | 7 | https://mathoverflow.net/users/nan | 218140 | 102,469 |
https://mathoverflow.net/questions/218058 | 12 | Let $M\_1$ and $M\_2$ be connected smooth manifolds and let $f\_0,f\_1:M\_1 \rightarrow M\_2$ be homotopic smooth maps such that some fixed point $p \in M\_2$ is a regular value for both $f\_0$ and $f\_1$. Question: Can I always find a smooth homotopy $F:M\_1 \times I \rightarrow M\_2$ such that $p$ is a regular value ... | https://mathoverflow.net/users/80162 | Homotopies with prescribed regular values | The answer is "yes" if $M\_1$ is compact. Here's a sketch of a proof.
Consider any smooth homotopy $G:M\_1 \times I \rightarrow M\_2$ between $f\_0$ and $f\_1$. Since $M\_1$ is compact, the set of regular values of $f\_0$ and $f\_1$ are both open. Moreover, the condition of being a regular value is stable. We can thu... | 6 | https://mathoverflow.net/users/317 | 218142 | 102,471 |
https://mathoverflow.net/questions/218095 | 7 | I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of codimension $1$). We know that we have an exact sequence, $$H^0(\mathcal{O}\_U) \to H^0(U-Z,\mathcal{O}\_U|\_{U-Z}) \xrightarr... | https://mathoverflow.net/users/45397 | Local cohomology groups and linearity | Please compute through an example before asking! What happens if $U = \text{Spec}(k[x])$ and $Z = V(x)$?
Later edit. The answer to your question is that if $A$ is a ring and $f \in A$ an element, and $M$ an $A$-module, then we can look at the sequence
$$
0 \to M[f^\infty] \to M \to M\_f \to M\_f/M \to 0 \to 0 \to \ld... | 4 | https://mathoverflow.net/users/80000 | 218149 | 102,472 |
https://mathoverflow.net/questions/218112 | 5 | So far I have seen the use of vector spaces in control theory and other notions from linear algebra; So I wonder if there's a use of this abstraction of modules over rings in control theory? any literature you can suggest?
Thanks.
| https://mathoverflow.net/users/13904 | The use of modules in control theory | Modules theory over the rings of principal ideals is the main tool of control theory.
Linear control theory mainly deals with matrices whose entries are polynomials.
Polynomials is the ring of principal ideals. Matrices over this ring require modules theory.
This fact is usually hidden in the books of control theory wr... | 7 | https://mathoverflow.net/users/25510 | 218150 | 102,473 |
https://mathoverflow.net/questions/218138 | 2 | Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is positive. Consider a pants decomposition containing $x$. Consider the Fenchel-Nielsen coordinate of the Teichmuller space with ... | https://mathoverflow.net/users/9485 | Is the length function associated with the twist parameter an increasing function? | Yes, this is shown in Kerckhoff's paper:
The Nielsen Realization Problem
Steven P. Kerckhoff
Annals of Mathematics
Second Series, Vol. 117, No. 2 (Mar., 1983), pp. 235-265
Proposition 3.5
| 3 | https://mathoverflow.net/users/11142 | 218153 | 102,475 |
https://mathoverflow.net/questions/218160 | 6 | In a paper that I am reading there is a following step:
>
> Let $X$ be a Banach space and let $(x\_k) \subset X$ be a normalized sequence that converges weakly to $0$.
> Then $\overline{co}(x\_k)$ is a weakly compact set.
>
>
>
(notice that $\overline{co}(x\_k)$ denotes the norm-closure of the convex hull of ... | https://mathoverflow.net/users/50818 | Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact? | You can find in many functional analysis text books the theorem that the closed convex hull of a weakly compact subset of a Banach space is weakly compact. But what you want is simpler than the general theorem. Here is a simple conceptual proof: Let $(y\_n)$ be a weakly null sequence in $X$ and consider the bounded lin... | 8 | https://mathoverflow.net/users/2554 | 218164 | 102,478 |
https://mathoverflow.net/questions/218165 | 3 | Suppose that we have a continuous open and closed surjection $f\colon X\to Y$ of a Polish space $X$ to $Y.$ The closeness of $f$ implies that $Y$ is a metric space.
But i do not know how to use that $f$ is continuous and open to prove that $Y$ is Polish, is there some result that implies this?
| https://mathoverflow.net/users/26901 | Continuous and open image of a Polish space | The following list of results will show that $Y$ is Polish.
* It is basic that the continuous image of a separable space is separable.
* Also basic is that the open image of a first-countable space is first-countable.
* Recall the Hanai-Morita-Stone Theorem:
>
> Let $X$ be a metrizable space, and let $f : X \to Y... | 6 | https://mathoverflow.net/users/13653 | 218179 | 102,483 |
https://mathoverflow.net/questions/217789 | 3 | It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on.
Now, I stumbled over the question whether in some particular cases, as I will describe below, local symplectomorphisms are indeed global ones.
The question arouse in the context of action-angle variables. I was re... | https://mathoverflow.net/users/77929 | Local symplectomorphisms become global ones? | Local symplectomorphism, in general, do not induce global ones. In fact every symplectic manifold is locally symplectomorphic to to the standard $(\mathbb R^{2n},\omega\_{n})$ (Darboux theorem) while not (even if diffeomorphic to it) not globally symplectomorphic to it.
Even in the context of action-angle variables o... | 4 | https://mathoverflow.net/users/6032 | 218185 | 102,486 |
https://mathoverflow.net/questions/218148 | 4 | I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or $H\mathbb{Z}$-one? how can one denote it?) is zero (in all degrees). My questions are:
1) Is it ok to call these spectra acycli... | https://mathoverflow.net/users/2191 | Does the (singular)cohomology of any acyclic spectrum vanish? | Let me address what hasn't been answered in comments (not in an optimal way, though).
1) is OK and, modulo the meaning of your quotation marks, the answer to 2) is 'no'. I mean, don't expect anything very explicit or much beyond the very definition, it's a very complicated problem. As for 5), the right othogonal is ... | 5 | https://mathoverflow.net/users/12166 | 218186 | 102,487 |
https://mathoverflow.net/questions/218171 | 3 | Recently, I read a letter, containing the following identity:
$$
\sum \_{q=-\infty }^{\infty } \frac{(-1)^q I\_q\left(\left| \alpha \right| ^2\right) I\_q\left(\left| \alpha \right| ^2\right)}{2 q+1}=\frac{\sinh \left(2 \left| \alpha \right| ^2\right)}{2 \left| \alpha \right| ^2}.
$$
The letter's author referred to th... | https://mathoverflow.net/users/80212 | The Identity of the Modified Bessel Function of the first kind | I don't see a relation between the two. But your first identity can be deduced from the relation
$$
\sum\_{n=-\infty}^\infty \frac{J\_n(y)^2}{n+x}=\frac\pi{\sin(\pi x)}J\_x(y)J\_{-x}(y)
\tag{2.3}
$$
established by [M. D. Rogers (2005)](http://www.ams.org/mathscinet-getitem?mr=2131261) (his equation numbering). In fact,... | 5 | https://mathoverflow.net/users/19276 | 218191 | 102,489 |
https://mathoverflow.net/questions/218192 | 16 | To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally available.
So, just out of curiosity, I would like to know, whether my impression is wrong and what current hot resear... | https://mathoverflow.net/users/31310 | Current Research in Numeric Mathematics | No, research in numerical mathematics is still very relevant today.
One of the main challenges is **big data**: scaling the usual algorithms up to larger dimensions. Today's linear systems may involve sparse matrices of dimensions 100k or 1M, for instance. Using traditional methods such as Gaussian elimination will t... | 37 | https://mathoverflow.net/users/1898 | 218195 | 102,490 |
https://mathoverflow.net/questions/218078 | 18 | I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum with respect to some homology theory $E$) and the values of the zeta function at some integers.
$$ |\pi\_{i}^{s}L\_{E}\ma... | https://mathoverflow.net/users/61328 | stable homotopy groups and zeta function | Here is a slightly more fleshed out version of my comment. Let $K(1)$ be the first [Morava $K$-theory](http://ncatlab.org/nlab/show/Morava+K-theory). When $p$ is odd one can calculate the homotopy groups of the $K(1)$-localised sphere spectrum to be
$$
\pi\_nL\_{K(1)}\mathbb{S} = \begin{cases}
\mathbb{Z}\_p, &n=0,1\\
... | 12 | https://mathoverflow.net/users/16785 | 218204 | 102,494 |
https://mathoverflow.net/questions/218194 | 7 | Let $f: X \dashrightarrow Y$ be a rational map where $X$ and $Y$ are varieties over $\mathbb{C}$ with $X$ smooth and $Y$ proper (or projective if necessary). Choose an open set $U \subset X$ on which $f$ is defined.
Pick a closed point $x \in X$ and an algebraic curve $C$ passing through $x$ such that $C \cap U \neq ... | https://mathoverflow.net/users/45609 | Does a rational function extend to a point if all its 1-parameter extensions at that point agree? | Thanks to Jason Starr who answered my question in the comments. I would like to beef up his answer here for future reference.
>
> The answer to the first question--and thus for the second--is in the affirmative. In fact, there are no restrictions on the singularities of $X$; for instance it need not be normal let a... | 4 | https://mathoverflow.net/users/45609 | 218212 | 102,497 |
https://mathoverflow.net/questions/218159 | 7 | Let $M$ be a real, even dimensional, compact manifold endowed with a symplectic form $\omega$ and a flat, torsionless connection $\nabla$ compatible with $\omega$, that is $$\nabla \omega=0.$$
Under this assumptions, taking into account a simplified version of the Markus conjecture, it should follow that $M$ is geode... | https://mathoverflow.net/users/70498 | Are compact, complex, affinely flat manifolds geodesically complete? | Now that you added the volume requirement, you get close to the Marcus conjecture. In complex dimension 1 you then get a locally Euclidean structure and the answer is clearly positive. In complex dimension 2 it is again positive since linear holonomy then has rank 1 and in this situation Marcus conjecture was proven by... | 4 | https://mathoverflow.net/users/21684 | 218217 | 102,501 |
https://mathoverflow.net/questions/215722 | 2 | Usually whenever one reads the definition of profinite group, one starts with an ordered set $I$ which is *directed*, meaning that for every $i,j\in I$ there is some $k\in I$ such that $i\leq k$ and $j\leq k$. Then one has a family of finite groups $\{G\_i\}\_{i\in i}$, compatible maps between them and defines $\varpro... | https://mathoverflow.net/users/36370 | Profinite groups, directed sets and $H^1$ | I hope this will clarify the situation:
I: Why is $I$ a directed set: In fact, there is no need for $I$ to be directed. One can define the inverse limit of a inverse system without the restriction that $I$ is directed. However, one should be aware that by doing so, certain propositions about profinite groups resp. sp... | 1 | https://mathoverflow.net/users/75418 | 218218 | 102,502 |
https://mathoverflow.net/questions/218211 | 3 | I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider the singular homology of the spectrum of a space it will be put in non-positive degrees). My (silly) question is: can one s... | https://mathoverflow.net/users/2191 | Is the "inverse" (i.e., the "cohomological") numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra "acceptable"? | I'd say it is an unfortunate accident you did that once, and you should not do it again. The question is not mathematics but readability: not a good idea to go against a universally accepted convention. The textbook Hilton and Wylie tried to go against convention (talking of contrahomology instead of cohomology) and pr... | 11 | https://mathoverflow.net/users/14447 | 218221 | 102,503 |
https://mathoverflow.net/questions/218207 | 27 | Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or Sullivan's classic textbooks - that last three of which I have read.
But what information about a space is contained in it... | https://mathoverflow.net/users/54788 | Why study the p-completions of a space? | First one should separate between the property and being $p$-complete and process of $p$-completion. In the classical setting, the $p$-completion functor is not so well-behaved for general spaces. For example, the $p$-completion of a space need not be $p$-complete. One way to remedy this is to notice that $p$-completio... | 25 | https://mathoverflow.net/users/51164 | 218224 | 102,504 |
https://mathoverflow.net/questions/218209 | 6 | I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial\_t \psi(x,t) = H\_0(x)\psi(x,t) + u(t)H\_1(x)\psi(x,t)$ be a Schrödinger equation. $H\_0,H\_1$ are nice operators (as nice as there is theory avai... | https://mathoverflow.net/users/77929 | Quantum Mechanics and bilinear optimal control theory | The earliest reference is [On the controllability of quantum‐mechanical systems](http://scitation.aip.org/content/aip/journal/jmp/24/11/10.1063/1.525634) (1983). For recent developments, see [On the problem of quantum control in infinite dimensions](http://arxiv.org/abs/1004.3447) (2011) and [Finite Controllability of ... | 3 | https://mathoverflow.net/users/11260 | 218228 | 102,505 |
https://mathoverflow.net/questions/163423 | 5 | I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the algebraic and geometric structures in this context. And this is the topic of this post.
Let $\ \*\ $ stand for a single-p... | https://mathoverflow.net/users/8385 | Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-) | It seems that the conjecture (H2) can be confirmed with help of the recent result of Hofmann and Kramer (<http://arxiv.org/pdf/1301.5114.pdf>) who proved that for a compact topological group $G$ and a closed subgroup $H\subset G$ the homogeneous space $X=G/H$ is a manifold if and only if $X$ contains a non-empty open s... | 9 | https://mathoverflow.net/users/61536 | 218230 | 102,506 |
https://mathoverflow.net/questions/218210 | 8 | Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients in $k$ (viewed as a trivial module), i.e.,
$$H(G):=\begin{cases}H^\*(G,k)&p=2\\H^{ev}(G,k)&p>2\end{cases}$$
For wha... | https://mathoverflow.net/users/32261 | Computations in modular cohomology of finite groups | I'll discuss $H^\*(GL\_n\mathbb{F}\_q; k)$ first, because that is my current area of research.
1) When $\mathbb{F}\_q$ and $k$ have different characteristics (although $p$ typically still divides the order of $GL\_n\mathbb{F}\_q$), the answer is completely computed by Quillen, "On the Cohomology and K-Theory of the G... | 18 | https://mathoverflow.net/users/5762 | 218236 | 102,510 |
https://mathoverflow.net/questions/130979 | 8 | **Background and motivation:**
Consider the cone $C\subset \mathbb{R}^d$ of vectors with non-negative components, and let $\Delta\subset C$ be the simplex of probability vectors (those for which $\sum v\_i = 1$). The cone (and hence the simplex) can be equipped with the [Hilbert metric](http://en.wikipedia.org/wiki/H... | https://mathoverflow.net/users/5701 | Hilbert metric and cross-ratio of points on simplices | The following is taken from *On convex projective manifolds and cusps* Adv. Math. 277 (2015), 181–251.
If Ω is properly convex, a function $f:\Omega\to{\mathbb R}$ satisfies the *maximum principle* if for every compact subset $K\subset\Omega$ the restriction $f|K$ attains its maximum at an extreme point of $K$.
Coroll... | 6 | https://mathoverflow.net/users/42506 | 218246 | 102,514 |
https://mathoverflow.net/questions/218238 | 5 | I am interested in whether each component of a divergenceless vector can itself be written as a divergence. My motivation for this question is the characterization of so-called *trivial conservation laws* in physics. I suspect this representation is general, but I have not made much progress in a proof.
**Overview**
... | https://mathoverflow.net/users/80241 | General solution to null-divergence equation | The answer to your question is 'yes', that is the general solution. This is one of the basic results in the theory of the variational bicomplex. It is a statement of the vanishing of a certain cohomology group in the variational bicomplex. A good place to look for the proof would be in say, these [lectures](http://digi... | 7 | https://mathoverflow.net/users/13972 | 218260 | 102,517 |
https://mathoverflow.net/questions/218271 | 1 | In his answer to user42090's mathoverflow question"Minimal Generalized Contnuum Hypothesis & Axiom of Choice", Prof. Hamkins writes:
"...one can build the analogue of the symmetric models for $\lnot$$AC$ above any cardinal, while preserving $GCH$ below..."
Are there any examples of such models in the literature whi... | https://mathoverflow.net/users/20597 | A question regarding models of $ZF+I_0$ [Revised] | I think there's a misunderstanding here: there is *one* method for building a model in which choice breaks, and the point is that to get a model in which choice fails and some large cardinal axiom $(\*)$ holds, we start with a model in which $\kappa$ has $(\*)$, and then cause a failure of AC sufficiently far above $\k... | 5 | https://mathoverflow.net/users/8133 | 218287 | 102,522 |
https://mathoverflow.net/questions/217776 | 5 | Let $F\_2$ be the free group on two generators.
Let $U\le F\_2$ be a characteristic subgroup of finite index, and let $f : F\_2\rightarrow\mathbb{Z}^2$ be the abelianization map.
It's easy to check that $f(U)$ is characteristic in $\mathbb{Z}^2$, so it must be of the form $N\mathbb{Z}\times N\mathbb{Z}$ for some $N... | https://mathoverflow.net/users/15242 | Abelianization of characteristic quotients of $F_2$ | Note that $N$ is clearly the exponent of $F\_2/[F\_2,F\_2]U$.
For any normal subgroup $U$ of $F\_2$, the exponent of $F\_2/[F\_2,F\_2]U$ is the same as the exponent of $K/[K,K]$, where $K=F\_2/U$. Thus, $N$ is the exponent of $K/[K,K]$.
Now, $K$ is a subgroup of the Cartesian product $G\times G\times\dots$ such tha... | 2 | https://mathoverflow.net/users/24165 | 218293 | 102,525 |
https://mathoverflow.net/questions/218289 | 6 | Does anyone know of the mean value of two Ramanujan Sums when summed over the square of integers?
In my research on the [Landau problem](https://en.wikipedia.org/wiki/Landau's_problems) regarding nearly square primes, I have run into the mean value of a product of [Ramanujan Sums](https://en.wikipedia.org/wiki/Ramanu... | https://mathoverflow.net/users/65913 | What is the mean value of a pair of Ramanujan Sums when summed over squares? | Here is a partial answer (**Added:** completed below). Assuming $(r,s)=1$ and denoting $q=rs$, we have
$$ \sum\_{n=1}^N c\_r\left( n^2 \right) c\_s\left( n^2 \right)
= \sum\_{n=1}^N c\_q\left( n^2 \right) = \sum\_{n=1}^N \sum\_{d\mid(q,n^2)}\mu\left(\frac{q}{d}\right)d = \sum\_{d\mid q}\mu\left(\frac{q}{d}\right)d\sum... | 9 | https://mathoverflow.net/users/11919 | 218296 | 102,528 |
https://mathoverflow.net/questions/218253 | 6 | Let $\mathfrak{g}$ be a semi-simple Lie algebra.
So in characteristic $0$, the Grothendieck group of a block of category $\mathcal{O}$ is given by the classes of the Verma modules. Unlike the simples (or projectives), Verma modules are very easy to understand explicitly, and one can easily perform computations with ... | https://mathoverflow.net/users/2623 | Well-understood bases for Grothendieck groups of modular representation categories | I'm not sure what sources you are mainly relying on, but there are several points to be made:
1) When you say the Lie algebra is "semisimple", I suspect you mean (as people sometimes do when using shorthand) the Lie algebra of a semisimple algebraic group. There are lots of other simple Lie algebras (mostly classifie... | 2 | https://mathoverflow.net/users/4231 | 218299 | 102,530 |
https://mathoverflow.net/questions/218259 | 6 | Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along $c$ leaves $K$ fibred with $F$ still being the fiber surface?
| https://mathoverflow.net/users/27433 | Can one twist fibred knots and still get fibred knots? | Such examples were [constructed by Morton](http://www.ams.org/mathscinet-getitem?mr=728587). He showed that one can find unknotted curves lying on fiber surfaces *with zero framing*. Twisting about them preserves fiberedness and the fact that it is a knot in $S^3$. Also, curves on the fiber can be pushed disjoint from ... | 10 | https://mathoverflow.net/users/1345 | 218302 | 102,532 |
https://mathoverflow.net/questions/218268 | 6 | For $k\in\mathbb{N}\_{0}$ and $x\in\mathbb{R}$, define
$$I\_{k}(x):=\int\_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$
where
$$g(\theta)=\int\_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-\alpha^{2}t^{2})}}$$
and $0<\alpha<1$ is a fixed parameter.
I arrived at the expression $I\_{k}(x)$ whil... | https://mathoverflow.net/users/56553 | Asymptotic behaviour of an integral | With the aid of Mathematica I found that
$$g(\theta)=\textrm{EllipticK}(a^2)-\textrm{EllipticF}(t,a^2).$$
I get the first terms of the asymptotic expansion
$$\frac{\sqrt{\pi}}{2\sqrt{k}}-\frac{\sqrt{\pi} x^2}{8(1-a^2)k^{3/2}}+
\Bigl(-\frac{k}{6}+\frac{a^2 x^2}{6(1-a^2)^2}+\frac{x^4}{24(1-a^2)^4}\Bigr)
\frac{3\sqrt{\pi... | 7 | https://mathoverflow.net/users/7402 | 218303 | 102,533 |
https://mathoverflow.net/questions/218311 | 12 | Let $M$ be a smooth compact Kahler manifold and let $\mathcal{F}$ be a local system on $M$.
Question 1: I assume that there exists a twisted Hodge to de Rham spectral sequence converging to $H^{p+q}(M;\mathcal{F})$ whose $E\_1$-page is of the form
$$E\_1^{p,q} = H^p(M;\mathcal{F} \otimes \Omega^q).$$
Is this corr... | https://mathoverflow.net/users/80162 | Hodge to de Rham spectal sequence with twisted coefficients | (Edit: I answered Q2 initially, ignoring Q1.)
Q1: The spectral sequence is right except that it starts at $E\_1$.
Q2: It is true if $\mathcal{F}$ is unitary in the sense that the underlying representation of $\pi\_1(M)$ is unitary, then the spectral sequence degenerates at the $E\_1$ page. This seems like a folklor... | 9 | https://mathoverflow.net/users/4144 | 218312 | 102,538 |
https://mathoverflow.net/questions/218316 | 11 | Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep track of their provenance. The nice thing about the simplicial case is that in order to get the whole combinatorial structur... | https://mathoverflow.net/users/36414 | "Database" of simplicial polytopes/spheres | You can find Frank Lutz's lists of simplicial spheres (and other manifolds) [here](http://page.math.tu-berlin.de/~lutz/stellar/3-manifolds.html):
Let me shamelessly self-advertise my list of simplicial 4-polytopes with up to $10$ vertices and various families of neighborly polytopes [here](http://page.mi.fu-berlin.de... | 15 | https://mathoverflow.net/users/39495 | 218319 | 102,541 |
https://mathoverflow.net/questions/218265 | 3 | The following (easy) inequality seems to be quite useful. It also seems to me likely to be known, however I do not know of any reference - is it known?
Given random variables $Y\_1,\dots,Y\_n$ with $Y\_i\in[0,a\_i]$ for each $i$, where the $a\_i$ are positive real numbers, suppose that there exist a real $p\_1$ and f... | https://mathoverflow.net/users/36212 | Reference request: Hoeffding-type inequality | If you define $Z\_i=Y\_i−p\_i(Y\_1,…,Y\_{i−1})$ then the sum $Z\_1+…+Z\_n$ is a supermartingale and the probability of it being bigger then $t$ gives you what you wanted.
| 7 | https://mathoverflow.net/users/1061 | 218322 | 102,542 |
https://mathoverflow.net/questions/218313 | 15 | Let $X\_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S\_n$. It is well known that the total cohomology of $X\_n$ is isomorphic to the regular representation of $S\_n$, but I would like to know what representation one gets in each degree.
Wh... | https://mathoverflow.net/users/10273 | Cohomology of configuration space as a representation of the symmetric group | One can indeed modify the formulas in the paper of Getzler to get an answer for any $\mathbf R^d$, by judiciously inserting minus signs and making substitutions $x \mapsto x^{d-1}$ in various places, but if I try I'll probably get it wrong. So let me explain how it works instead, and hopefully there are no sign errors ... | 18 | https://mathoverflow.net/users/1310 | 218336 | 102,544 |
https://mathoverflow.net/questions/218283 | 66 | The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the K3 surface from a algebraic topology perspective (hence I view it as a particular smooth 4-manifold).
As we have seen... | https://mathoverflow.net/users/184 | Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres? | Yes, such M exists. The boundary connected sum of 28 copies of the Milnor plumbing has boundary diffeomorphic to $S^7$ so it can be closed off with $D^8$ and you can let $M$ be the connected sum of the resulting closed manifold and 8 copies 7 copies of $S^4 \times S^4$.
Why does that work? Let $f: M \to \mathbb{O}P^1... | 44 | https://mathoverflow.net/users/80296 | 218341 | 102,547 |
https://mathoverflow.net/questions/218135 | 8 | Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map $P\otimes\_A P\to P\otimes H$ is a 1-1 correspondence, where $A$ is the $H$ invariant part of $P$.
There is an averaging... | https://mathoverflow.net/users/29625 | Hopf Galois extensions and conditional expectations for C* algebras | First off, as you can see, the definition of $E$ doesn't require the Hopf-Galois condition. The positivity is a consequence of that for slicing by states. If $\phi$ is a state on a C$^\*$-algebra $C$, the map $\iota\otimes\phi\colon B \otimes\_{\rm min} C \to B$ characterized by $b \otimes c \mapsto \phi(c) b$ is (comp... | 3 | https://mathoverflow.net/users/9942 | 218349 | 102,550 |
https://mathoverflow.net/questions/218347 | 2 | Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be
$$
B(X,k)=\{(x\_1,x\_2,\cdots,x\_k)\in X^k\mid x\_i\neq x\_j \text{ for any } i\neq j\}
$$
and the relative unordered configuration space be
$$
B(X,A;k)=\{(x\_1,x\_2,\cdots,x\_k)\in X^k\mid x\_i\neq x\_j ... | https://mathoverflow.net/users/65800 | distinct multiple points in a space with at least one point lying in a subspace | You have to map $B(A,j)\times B(X\setminus A,k-j)$ to $B(X,A;k)$ by the "union" map $\phi:2^X\times 2^X \to 2^X$ defined as $\phi(C,D)=C\cup D$. Also define $B(Y,n)$ for $n\ge 0$, not just $n\ge 2$. Then $B(X,A;k)=\cup\_{j=1}^k \phi(B(A,j)\times B(X\setminus A,k-j))=\cup\_{j=1}^k\{\{x\_1,\cdots,x\_k\}:\{x\_1,\cdots,x\_... | 0 | https://mathoverflow.net/users/75422 | 218352 | 102,551 |
https://mathoverflow.net/questions/218309 | 7 | Let $\|M\|:=\sup\_{u:\|u\|=1}\|Mu\|$ be the operator norm induced by the Euclidean distance.
Suppose $A$ is a $k\times k$ symmetric matrix with $A\_{ij}>0$ for all $i,j$ and $\sum\_{i,j} A\_{ij} = 1.$ Let $A\_i$ be the sum of all elements in row $i$ (or column $i$) of $A.$ Let $B$ be a matrix with entries $$B\_{ij} ... | https://mathoverflow.net/users/7576 | Approximation theoretic question about operator norm | I discovered that my conjecture is incorrect. Here is a simple counterexample.
First, let's allow $A\_{ij}=0$ for some $i,j.$ Let
$$A\_{ij} = \begin{cases}\frac{1-\epsilon}{(k-1)^2} & 1\leq i,j\leq k-1 \\
\epsilon & i=j=k \\
0 & \mbox{else}.
\end{cases}$$
Then, $\|B\| = 1$ since $B\_{kk} = 1,$ and we know that $\... | 0 | https://mathoverflow.net/users/7576 | 218367 | 102,556 |
https://mathoverflow.net/questions/218361 | 6 | [This question is an extension of my question [Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of $\mathbb{R}$?](https://mathoverflow.net/questions/210316/does-a-positive-measure-subset-of-the-unit-interval-almost-surely-intersect-a-ra). I'm as... | https://mathoverflow.net/users/15570 | Can the integral of a "generic" bounded measurable function be determined by its values on the rationals? | Yes your $F$ works. This is a result of B. Jessen, "On the Approximation of Lebesgue Integrals by Riemann Sums", Annals of Mathematics Second Series, Vol. 35, No. 2 (Apr., 1934), pp. 248-251.
I found this by googling "Strong Sweeping Out" and "Riemann sums". If you do the Riemann sum along multiples of $1/n$, however... | 7 | https://mathoverflow.net/users/11054 | 218370 | 102,558 |
https://mathoverflow.net/questions/218372 | 12 | When I wrote my master's thesis, a professor who read it said that I should not use the phrase "A function of class $k$." but instead "A function of class $C^k$". I am not an expert about mathematical history of notations, but I read that in Geometric Measure Theory, H. Federer actually uses the first one, and it seems... | https://mathoverflow.net/users/56191 | Mathematical writing : using an "out-of-date" notation | Federer was not exactly known, even to his contemporaries, for employing standard notation. Here is a quote from Steenrod's 1948 [Math Review](http://www.ams.org/mathscinet-getitem?mr=27161) of some mimeographed notes of Federer for a course on differential geometry.
>
> The most striking feature of the book to the... | 55 | https://mathoverflow.net/users/66607 | 218380 | 102,562 |
https://mathoverflow.net/questions/202513 | 2 | **Context**
Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}\_{q}\left(\mathcal{L}\mathfrak{sl}\_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the Drinfeld generator $x\_{m}^{+}$ ($x^{-}\_{m}$) acts by $k^{m}X$ ($k^{m}Y$) on $V$.
We recall that $\mathcal{U}\_{q}\le... | https://mathoverflow.net/users/57464 | How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module? | For the meaning of $T(z)$, see the paper of Frenkel and Hernandez (<http://arxiv.org/abs/1308.3444>, version 4) Example 5.6. The eigenvalues of $T(z)$ are related to the so-called Frenkel-Reshetikhin q-character of $V$. See Proposition 5.8 of that paper for the precise statement and the proof.
| 3 | https://mathoverflow.net/users/80317 | 218381 | 102,563 |
https://mathoverflow.net/questions/218378 | 1 | I'm beginning my university course in Theoretical Physics next week and we have been asked to choose our modules.
We have compulsory modules in Physics, Vector Algebra and Dynamics. But we also have the choice to study some Pure Math modules (Numbers, Sets and Sequences then Linear Algebra) or Computational Physics mo... | https://mathoverflow.net/users/80315 | Computational Physics or Pure Math Modules for Theoretical Physics | The answer is unfortunately subjective. When I started with down the path of theoretical physics, I would have chosen the computational side - it's certainly useful throughout theoretical physics.
On the other hand, the mathematical side is much more interesting to me. As I had a decently strong background in compute... | 1 | https://mathoverflow.net/users/43265 | 218384 | 102,564 |
https://mathoverflow.net/questions/218344 | 5 | I have a random variable $X$ whose first and second moments are given as
$$
E[X] \propto C\_n^{1-a},\quad E[X^2] \propto C\_n^{2-a}\quad (0 < a < 1),
$$
where $C\_n$ satisfies
$$
\lim\_{n\to\infty} C\_n = \infty, \quad \lim\_{n\to\infty} \frac{C\_n^a}{n} = 0.
$$
I want to see the limit of a ratio,
$$
R\_n = \frac{\sum... | https://mathoverflow.net/users/80302 | The law of large numbers for diverging moments | I interpret the question as follows (cf. the comment by **Nate Eldredge**).
For each natural $n$, let $X\_n,X\_{n,1},\dots,X\_{n,n}$ be independent identically distributed (i.i.d.) random variables (r.v.'s) such that
$$
EX\_n \bowtie C\_n^{1-a},\quad EX\_n^2\bowtie C\_n^{2-a}\quad (0 < a < 1),
$$
where $C\_n$ satis... | 4 | https://mathoverflow.net/users/36721 | 218386 | 102,565 |
https://mathoverflow.net/questions/218335 | 2 | Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma\_n$ act on
$$
\prod\_n X
$$
by
$$
\sigma(x\_1,\cdots,x\_n)=(x\_{\sigma(1)},\cdots,x\_{\sigma(n)}), \sigma\in \Sigma\_n.
$$
Then we have a quotient space
$$
(\prod\_n X)/\Sigma\_n.
$$
By Kunneth formula,
$$
H^\*(\prod\_nX; F)=... | https://mathoverflow.net/users/65800 | Kunneth formula of Cartesian product modulo orders of coordinates | Firstly, your space $(\prod\_n X)/\Sigma\_n$ is better known as the $n$-fold symmetric product of $X$, and denoted $SP^n(X)$. [Milgram](http://www.maths.ed.ac.uk/~aar/papers/milgram4.pdf) (building on work of Steenrod, Cartan, Dold, Nakoaka and others) has shown how to compute $H\_\ast(SP^n(X);k)$ for any field $k$ and... | 2 | https://mathoverflow.net/users/8103 | 218388 | 102,566 |
https://mathoverflow.net/questions/210821 | 6 | Suppose that $\lambda$ is a cardinal. Let $\mathcal{E}\_{\lambda}$ be the set of all elementary embeddings from $V\_{\lambda}$ to $V\_{\lambda}$. If $j,k\in\mathcal{E}\_{\lambda}$, then define
$j[k]=\bigcup\_{\alpha<\lambda}j(k|\_{V\_{\alpha}})$. Suppose that $j\_{1},...,j\_{n}\in\mathcal{E}\_{\lambda}$. Let $\langle j... | https://mathoverflow.net/users/22277 | Does the critical sequence for subalgebras of elementary embeddings with finitely many generators have order type $\omega$? | I claim that the answer to both questions is **yes**
.
Suppose that $A$ is a finite set, $j\_{a}\in\mathcal{E}\_{\lambda}$ for each $a\in A$ and $\gamma<\lambda$ is a limit ordinal. Then I claim that $\langle\{j\_{a}|a\in A\}\rangle/\equiv^{\gamma}$ is finite and the sequence $\{\textrm{crit}(j)|j\in\langle\{j\_{a}|a\i... | 3 | https://mathoverflow.net/users/22277 | 218395 | 102,568 |
https://mathoverflow.net/questions/218292 | 6 | I'm sure this must be covered somewhere, but all the references I have only treat this in very special cases (mostly when working over fields).
Suppose $f : X\rightarrow S$ is smooth of finite presentation with geometrically connected fibers of dimension 1, and let $g\_i : S\rightarrow X$ be finitely many sections. L... | https://mathoverflow.net/users/15242 | Can you functorially "reconstruct" a branched cover of curves from its etale locus? | 2) Yes, if the cover is finite etale. (Maybe we need to be over a Noetherian base as well?) All the functions on the cover are going to be integral over the base and contained in the field of fractions of the open subset. Furthermore, there will be no additional functions of this type in the field of fractions of the c... | 4 | https://mathoverflow.net/users/18060 | 218405 | 102,575 |
https://mathoverflow.net/questions/218351 | 2 | Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a smooth, Riemannian manifold admitting a left action of $G$. Denote, for simplicity, the associated bundle by $E(M):= Q\times\_{G}M$. Then $\G... | https://mathoverflow.net/users/80304 | Sobolev Multiplication theorem for Fibre bundles | I have a solution in mind. But for that the embedding $E(M) \to \mathbb R^N$ is unnatural. So, I proceed in a slightly different way. Firstly, in any case it is required that $kp>4$, because the problem will involve left composing a Sobolev function with a smooth function. These operations are well-behaved only above t... | 2 | https://mathoverflow.net/users/15197 | 218417 | 102,580 |
https://mathoverflow.net/questions/218383 | 0 | Consider a twice differentiable strongly convex function $f:\mathbb{R}^n \rightarrow \mathbb{R^+}$ that attains its minimum value at the point $x^\*$. I am wondering if one can compute a direction of *slowest* ascent $u$ from the point $x^\*$.
Intuitively, the direction of slowest ascent would be a unit vector $u$ s... | https://mathoverflow.net/users/49673 | How to compute the direction of slowest ascent from the minimum of a strongly convex function? | There may not be a unique 'direction of slowest ascent', take for example the simple case of $f(\vec{x}) = x\_0^2 + x\_1^2$, in which case the minimum is clearly (0,0), and any direction is equally fast. That said, near enough to the minimum, the you can define a 'subspace of slowest ascent' which is basically just the... | 1 | https://mathoverflow.net/users/14424 | 218418 | 102,581 |
https://mathoverflow.net/questions/218414 | 12 | My understanding is that large cardinals are ordered by "consistency strength", but how does this correlate with their size (cardinality)?
More specifically, are there any systematic results on the lines of:
If A and B are two types of large cardinals such that
Cons(ZFC + Type A exists) => Cons( ZFC + Type B exists)... | https://mathoverflow.net/users/76572 | Large cardinal consistency strength and size | I may note that a cardinal of type $A$ may has more consistency strength of a cardinal of type $B$, while the smallest cardinal of type $A$ is smaller than the least cardinal of type $B$ (assuming cardinals of both types exist). For example:
>
> The consistency of a huge cardinal implies the consistency of a superc... | 17 | https://mathoverflow.net/users/11115 | 218419 | 102,582 |
https://mathoverflow.net/questions/218220 | 3 | Let $J\_t$ be a standard Brownian motion, let $X = \{t : J\_t = 0\}$ denote the zero set, and let $I(j, n)$ denote the indicator function of the event$$\left\{\text{there exists }s \in \left[{{j-1}\over{n}}, {j\over{n}}\right] \text{ with }J\_s = 0\right\}.$$Let$$K\_n = \sum\_{j=1}^n I(j, n).$$Observe that $K\_n$ denot... | https://mathoverflow.net/users/nan | Standard Brownian motion, limit, square of expectation bound | (Since I get a slightly different constant than cardinal, I detail a bit the computation).
For the first part, you need to compute the probability of zero crossing in $[t,t+\Delta]$ for $\Delta=1/n$. This is (conditioning on the value $z$ at time $t$ and using the reflection principle)
$$A=2\int\_0^\infty dz \frac{e^... | 5 | https://mathoverflow.net/users/35520 | 218446 | 102,591 |
https://mathoverflow.net/questions/218443 | 1 | Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B\_i\}\_i$ be a finite family of balls covering $K$. One may assume that all balls have the same radius, but it might be unnecessary.
**Is it true that one can choose a subcovering such that ever... | https://mathoverflow.net/users/16183 | Multiplicity of a subcovering in spaces of given Hausdorff dimension | The answer is no.
Hausdorff dimension does not reflect any global geometry.
Say you can construct a metric graph which approximates any compact length-metric space
(as well as a finite metric space which approximates any compact metric space).
| 2 | https://mathoverflow.net/users/1441 | 218448 | 102,593 |
https://mathoverflow.net/questions/218375 | 4 | Suppose we have the following:
* A C\*-algebra $A$ and a von Neumann algebra $M$ (we can assume that $M$ is $\mathcal B(H)$).
* A sequence of \*-homomorphisms $\phi\_i\colon A\to M$
* an ultrafilter $\mathcal U\in\beta\mathbb N\setminus\mathbb N$
Define $\phi(a)=\lim\_{i\in\mathcal U}\phi\_i(a)$ for $a\in A$ where ... | https://mathoverflow.net/users/47948 | Point-ultraweak limit of *-homomorphisms/cpc order zero maps | No.
Set $M:=\mathcal B(\mathcal H)$ and let $(e\_j)$ be an ONB for $\mathcal H$.
Set
$$ S\_i(e\_j) := \begin{cases} e\_0, \quad &j=0; \\ -e\_0,\quad &i=j; \\ 0, \quad &\text{otherwise} \end{cases} $$
and
$$ T\_i(e\_j) := \begin{cases} e\_0+e\_i, \quad &j=0; \\ 0, \quad &\text{otherwise} \end{cases} $$
Then $S\_iT\_... | 6 | https://mathoverflow.net/users/22052 | 218455 | 102,594 |
https://mathoverflow.net/questions/214748 | 6 | Let $A \subset k[x\_1, \dots, x\_n]$ be a subalgebra, which is also a graded subspace $A = \oplus\_{i \ge 0} A\_i$. One can write $A = A\_0 \oplus A\_{> 0}$ where we have $A\_0 = k^0[x\_1, \dots, x\_n] = k$ and $A\_{> 0} := \oplus\_{i \ge 1} A\_i$ is a graded ideal of $A$, known as the augmentation ideal.
My question... | https://mathoverflow.net/users/nan | Augmentation ideal is finitely generated if and only if $A$ is finitely generated as a $k$-algebra? | This amounts to an exercise, applying what are perhaps the first few theorems you encounter in the context of noetherian rings. Moreover, the assumption that $A$ is a graded subring of $k[x\_1,\ldots,x\_n]$ is stronger than necessary. It is enough to assume that $k$ is a commutative noetherian ring and that $A$ is a co... | 2 | https://mathoverflow.net/users/7932 | 218461 | 102,595 |
https://mathoverflow.net/questions/218421 | 1 | Let $R=k[x\_1,\ldots,x\_n]$ be a graded ring and $S,T,U$ be monomials ideals.
$reg(S)=max\{j-i \backslash \beta\_{i,j}(S) \neq 0\}$.
Assume $S+T=U$
**prove \disprove :** $reg(S+T^2) \leq reg(U^2)$.
We can see $reg(S+T^2) \leq reg(S)+reg(T^2)-1 < reg(S)+reg(T^2)$ by [By Herzog result, see Corollary 3.2](http://arx... | https://mathoverflow.net/users/68302 | Castelnuovo- Mumford regularity properties | This example is due to Aldo Conca:
$R = k[x\_1,x\_2,x\_3], S = (x\_1^3x\_2,x\_1x\_2^3,x\_2^4,x\_1^2x\_2^2x\_3^5), T = (x\_1^4), U = S+T$. Then $\text{reg} (S+T^2) = 9$, while $\text{reg} U^2 = 8$.
| 3 | https://mathoverflow.net/users/43438 | 218462 | 102,596 |
https://mathoverflow.net/questions/218413 | 1 | I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading.
Consider two probability measures $P$ and $Q$ both absolutely continuous to a given measure. Then for any event $A$ we have,
\begin{align}
P(A) + Q(A^c) \geq \frac{1}{2}\exp\{... | https://mathoverflow.net/users/51716 | KL divergence Inequality | $\def\KL{\mathsf{KL}}$I'm not an expert (sorry), but it is intuitively obvious (and should follow from the standard properties) that $\KL(P,Q)$ would decrease if we replace $P$ and $Q$ by $\bar P$ and $\bar Q$ which are proportional on $A$ and $A^c$, and $\bar P(A)=P(A)$, $\bar Q(A)=Q(A)$.
If so, then the required in... | 3 | https://mathoverflow.net/users/17581 | 218466 | 102,598 |
https://mathoverflow.net/questions/217516 | 5 | I am reading Pinelis "[An approach to inequalities for the distributions of infinite -dimensional martingales](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)" and cannot follow his proof of Theorem 3:
Let $(f\_n)$ be a martingale in a separable Banach space $(\mathcal{X},||~||)$, $\mathcal{X} = ... | https://mathoverflow.net/users/42531 | Proof of Pinelis (1992) - Banach space inequalities | As written in my paper [[1](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)], the inequality
$$P(f^\*>r) \le 2\exp\big(-r^2/2(p-1)\big)
$$
in Theorem 3 in [[1](http://link.springer.com/chapter/10.1007/978-1-4612-0367-4_9#page-1)]
for martingales in $\mathcal{X}=L^p$ can be compared with the inequ... | 14 | https://mathoverflow.net/users/36721 | 218471 | 102,601 |
https://mathoverflow.net/questions/218422 | 9 | Is there a one-relator group with property (T)?
That is, is there an $n > 2$, and some $x \in F\_n$ (the free group on $n$ generators) such that the quotient of $F\_n$ by the normal subgroup generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?
| https://mathoverflow.net/users/38889 | Is there a one relator group with property (T)? | No, the abelianization of any such quotient will be infinite (the abelianization of $F\_n$ is $\mathbb{Z}^n$, which does not have a cyclic subgroup of finite index), but Kazhdan groups always have finite abelianization.
| 15 | https://mathoverflow.net/users/68305 | 218473 | 102,602 |
https://mathoverflow.net/questions/218484 | 4 | This may be trivial but I cannot find it. Given an elliptic curve, $C$ over $R$ with chosen parametrization $R\to \mathbb{A}^5\_{\mathbb{Z}}=Spec(A)$, is there a way to compute coefficients of the associated formal group law, assuming this is enough information to coordinatize it. I would assume that the coefficients a... | https://mathoverflow.net/users/41103 | Is it possible to compute coefficients of the formal group of an elliptic curve? | I assume you mean Weierstrass parametrization. Then $-x/y$ is the standard coordinate near 0, and the relevant formulas can be found in Silverman's book on elliptic curves.
| 6 | https://mathoverflow.net/users/80363 | 218486 | 102,605 |
https://mathoverflow.net/questions/4394 | 8 | A similar question reminds me: When giving talks, I often want to refer to the work of Henry Crapo. I have asked several mathematicians, and none of them were sure how to pronounce his last name. Any help?
| https://mathoverflow.net/users/297 | Pronunciation: Crapo | Gordon Royle is right, I'm living in La Vacquerie. The reference to GWU is not correct: that is the workplace of my colleague Bill Schmitt. The US pronunciation is indeed "cray-poe", but in France it tends to become "crah-poe".
Henry
| 38 | https://mathoverflow.net/users/80366 | 218491 | 102,607 |
https://mathoverflow.net/questions/218490 | 5 | Let $k$ be a commutative ring with unit and let $(A,\varepsilon)$ be a (not necessarily commutative) augmented, finitely generated $k$-algebra with augmentation ideal $I$.
If $\mu(A)$ denotes the minimal number of algebra generators of $A$ and $\mu(I)$ the minimal number of generators of $I$ as ideal, we have $\mu(I... | https://mathoverflow.net/users/17734 | Minimal number of algebra generators of a group ring | The paper [The presentation rank of a direct product of finite groups](http://www.sciencedirect.com/science/article/pii/0021869374900623) by Cossey, Gruenberg and Kovacs shows that the difference between the minimal number of generators of $G$ and $I$ can be as big as you like for finite groups.
| 6 | https://mathoverflow.net/users/15934 | 218494 | 102,609 |
https://mathoverflow.net/questions/218468 | 16 | Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C\_n$ and $H\_n$ of $R$-modules, which are tame in every possible sense -- finite rank, zero for large enough $n$, etc. Here's the problem:
>
> What is the set (space? category... | https://mathoverflow.net/users/18263 | Moduli space of boundary maps with prescribed chain and homology groups? | Over a field $k$, this is a special case of quiver loci. The general framework is that you are given a triangular array $(r\_{ij})\_{1 \leq i \leq j \leq n}$ of nonnegative integers. Consider a list of $n-1$ matrices $M\_1$, $M\_2$, …, $M\_{n-1}$ of size $r\_{ii} \times r\_{(i+1)(i+1)}$. The quiver locus is the set whe... | 7 | https://mathoverflow.net/users/297 | 218501 | 102,613 |
https://mathoverflow.net/questions/218481 | 8 | Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g\_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g\_t$ is smooth in time, or $k = \omega$, when the one-parameter family $g\_t$ is real analytic in time. Now, as the metric... | https://mathoverflow.net/users/65799 | $C^k$ one-parameter family of metrics | The references given in [this answer](https://mathoverflow.net/questions/179122/behavior-of-the-spectrum-of-the-laplacian-under-pointed-smooth-convergence/179776#179776) to a related issue provide at least a partial answer to your question.
On a *complete* Riemannian manifold $(M,g)$, the Laplacian associated with $... | 6 | https://mathoverflow.net/users/43324 | 218502 | 102,614 |
https://mathoverflow.net/questions/218507 | 6 | Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity **equally**. These $k$ particles move on $M$ freely without frictions and mutually repulse from each other. When these $k$ particles stop at $(x\_1,\cdot... | https://mathoverflow.net/users/80110 | electron configuration on manifolds | There seems to be a considerable body of work on this:
Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds
D.P. Hardin, , E.B. Saff1,
[More by Hardin and Saff.](http://www.math.vanderbilt.edu/~esaff/texts/201.pdf)
<http://personales.unican.es/beltranc/archivos/FoCMBeltran2011volu... | 6 | https://mathoverflow.net/users/11142 | 218524 | 102,619 |
https://mathoverflow.net/questions/218190 | 5 | Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index unital inclusion of hyperfinite ${\rm II}\_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M\_n)$ is depth $2$ (*reducible*) and is isomorphic to $(R^H \subset R)$ with $H$ a weak Kac algebra (also called quantum groupoid, see ... | https://mathoverflow.net/users/34538 | What are the applications of the depth 2 reduction to the subfactors theory? | *Two answers of Dmitri Nikshych:*
>
> **Dmitri**
>
> This reduces the study of subfactors to that of weak Hopf algebras.
> Equivalently (and more importantly) it relates the theory of finite
> index finite depth subfactors to the theory of fusion categories and
> module categories over them.
>
>
> There a... | 3 | https://mathoverflow.net/users/34538 | 218536 | 102,620 |
https://mathoverflow.net/questions/218531 | 7 | Recall the second Chebyshev function: $$\psi(x) = \sum\_{p \leq x} \lfloor \log\_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$.
In a hunt for an "elementary" lower bound on the number of primes up to some $x \in \mathbb{N}$, it is sufficient to find a lower bound for $\ps... | https://mathoverflow.net/users/38889 | An elementary lower bound on the number of primes | This question (with copious references, summarizing the progress to date, with best constants) is dealt with in [this 2013 paper by D. Bazzanella.](http://porto.polito.it/2517108/2/small_polynomials.pdf)
| 5 | https://mathoverflow.net/users/11142 | 218540 | 102,623 |
https://mathoverflow.net/questions/168882 | 3 | There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of [Grigor'yan-Nadirashvili-Sire](http://arxiv.org/abs/1406.0317) and references therein. I wonder how to estimate the smallest eigenvalue $\lambda\_1$ of a Schrodinger operator $-\Delta+h$ on a... | https://mathoverflow.net/users/38600 | Estimate the smallest eigenvalue of a Schrodinger operator | This question is actually quite subtle (at least I think so). The lowest eigenvalue of the Schrodinger operator $-\Delta\_g+h$ on a closed Riemannian manifold, where the potential $h$ is as you described, can actually be positive, negative or zero, and the sign depends upon the "size" of the potential.
I'll say more ... | 6 | https://mathoverflow.net/users/80390 | 218568 | 102,632 |
https://mathoverflow.net/questions/218550 | 11 | I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably correspond to the study of $p$-torsion of (co)homology. As far as I remember, $S^0/p$ is not a ring spectrum by a resul... | https://mathoverflow.net/users/2191 | Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$? | Depending on exactly what you mean by "killed by $p$", the answer may be no.
Let $\mathcal{C}$ be a stable $\infty$-category and let $\iota\_{\mathcal{C}}$ be the identity functor from $\mathcal{C}$ to itself. If the ``multiplication by $p$'' map
$p: \iota\_{\mathcal{C}} \rightarrow \iota\_{\mathcal{C}}$ is nullhomotop... | 15 | https://mathoverflow.net/users/7721 | 218569 | 102,633 |
https://mathoverflow.net/questions/218548 | 6 | For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ any root of unity (not necessarily related to $\zeta$).
>
> I have observed that **the minimal polynomial of $$(1+\zeta^n)\... | https://mathoverflow.net/users/29783 | Why are most coefficients of these minimal polynomials divisible by $p$? | So $\zeta=-\xi$, where $\xi$ is a primitive $p$'th root of unity. Let $\pi=1-\xi$. The minimal polynomial of $(1+\zeta^n)a$ is a product of terms of the form $X-(1+\zeta^{nj})\alpha\_{ij}$, where $j$ is odd and $i$ runs over some index set (maybe depending on $j$). Note that $j$ is odd because the Galois conjugates of ... | 8 | https://mathoverflow.net/users/11926 | 218571 | 102,634 |
https://mathoverflow.net/questions/218518 | 43 | Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can tell after some googling, a new family of integral representations of $\zeta(2n+1)$. I was told I should post this here b... | https://mathoverflow.net/users/80377 | Is this integral representation of $\zeta(2n+1)$ known? | As indicated in my comment, some of these integrals are essentially known, and involve the hyperbolic "Beukers-Kolk-Calabi" change of variables.
In particular, in [this paper](http://arxiv.org/pdf/1003.3602v1.pdf), Z. Silagadze shows in (27) that
\begin{equation\*}
\zeta(n) = \frac{2^n}{2^n-1}\int\_0^1\cdots\int\_0^... | 23 | https://mathoverflow.net/users/8430 | 218581 | 102,638 |
https://mathoverflow.net/questions/218270 | 26 | UPDATE: I am grateful to Peter May for the accepted answer, which makes most of the details below irrelevant. However, I will leave them in place for the record.
I am trying to understand the proof of Theorem 2 in the 1969 paper [Remarks on the Structure of Hopf Algebras](http://dx.doi.org/10.2307/2036615) by Peter M... | https://mathoverflow.net/users/10366 | Structure of Hopf algebras - trouble understanding an old paper | Ah, Neil, my apologies, sins of a forgetful old man. You are quite
right to object to Theorem 3 in that 1969 paper, because it is
false as stated. In fact, Paul Goerss gave me a counterexample ages ago. That appears as Example 23.4.2 in More Concise
Algebraic Topology, by Kate Ponto and myself, which is available
her... | 20 | https://mathoverflow.net/users/14447 | 218601 | 102,645 |
https://mathoverflow.net/questions/218566 | 1 | Suppose $D \subset \mathbb{R}^d$ is a domain and $f: \overline{D} \to \mathbb{R}$ is a continuous function, $C^2$ in $D$, satisfying$$f(x) = 0\text{ for }x\in \partial D,$$$${1\over2} \Delta f(x) = -1 \text{ for }x \in D.$$Let $B\_t$ be standard $d$-dimensional Brownian motion and let $\tau = \text{inf}\{t \ge 0 : B\_t... | https://mathoverflow.net/users/nan | $M_t = f(B_{t \wedge \tau}) + (t \wedge \tau)$ local martingale, $\textbf{E}^x[\tau] = f(x)?$ | Itô's formula asserts that $Y\_t = f(B\_t) + t$ is a (continuous) local martingale. Let $\tau\_n = \inf\{t : |Y\_t| \ge n\}$; then $\tau\_n \uparrow \infty$ and $Y\_{t \wedge \tau\_n}$ are bounded martingales. Then for each $n$, $M\_{t \wedge \tau\_n} = Y\_{(t \wedge \tau) \wedge \tau\_n}$ is a bounded martingale too. ... | 1 | https://mathoverflow.net/users/4832 | 218602 | 102,646 |
https://mathoverflow.net/questions/209140 | 47 | Let $m$ be any positive integer.
$$
P\_m(x)=\sum\_{i=0}^{m}\sum\_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.
$$
Question: $P\_m(x)$ always has integer values at all integers.
Some remarks:
(1) A polynomial is $\mathbb Z$-valued iff its unique expansion... | https://mathoverflow.net/users/6104 | How to prove this polynomial always has integer values at all integers? | $$P\_m(x)=\sum\_{i=0}^{m}\sum\_{j=0}^{m}\binom{x+j}{ j}\binom{x-1}{ j}\binom{j}{ i}\binom{m}{ i}\binom{i}{ m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$
Our task is to show it takes integer values on integers.
Folowing Wadim Zudilin we put
$$B\_k(x)=\binom{x+k}{2k}+\binom{-x+k}{2k}.$$
For $k\geq0$ the $B\_k$ are even po... | 35 | https://mathoverflow.net/users/4794 | 218606 | 102,648 |
https://mathoverflow.net/questions/217411 | 8 | An [addition chain](https://en.wikipedia.org/wiki/Addition_chain) for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for example, to calculate $x^n$ quickly by multiplying previously-calculated v... | https://mathoverflow.net/users/34461 | Upper bound on length of addition chain | Alfred Brauer, On addition chains,
Bull. Amer. Math. Soc. 45 (1939), 736-739
<http://www.ams.org/journals/bull/1939-45-10/S0002-9904-1939-07068-7/>
gives inequalities of the type you mention:
equation (11), ($\log n$ being the log with base e).
If $2^m < n<2^{m+1}$, then
$$l(n) \leq \min\_{1\leq r \leq m} \Bigl( (1... | 4 | https://mathoverflow.net/users/36707 | 218608 | 102,650 |
https://mathoverflow.net/questions/218604 | 10 | Is it true that the number of carries, when calculating the sum of a finite set of finite positive integers, is constant (i.e. independent of their permutation and the order in which the additions are carried out)? Carries are computed in base 2, so that 1+3 is $01\_2+11\_2=100\_2$, which involves 2 carries: the least ... | https://mathoverflow.net/users/31310 | Is the Number of Carries in Integer-Addition Associative? | In base $b$,
let $G\_n$ be the set of $n$-digit integers, thought of as the integers mod $b^n$. Then we have an exact sequence
$$0\rightarrow G\_1\rightarrow G\_n\rightarrow G\_{n-1}\rightarrow 0$$
For $n-1$ digit numbers, the leftmost
carry digit is the two-cocycle associated to this group extension and therefore ... | 10 | https://mathoverflow.net/users/10503 | 218619 | 102,655 |
https://mathoverflow.net/questions/218516 | 7 | I'm studying Serre's paper in wich he shows the following theorem:
*Let K be a number field, $E$ an elliptic curve over K without CM. Then the representation $$\rho\_{\ell}:\mathrm{Gal}(\bar K/K)\longrightarrow\mathrm{Aut}(E[\ell])$$ is surjective for all but finitely many prime numbers $\ell$.*
I see the beauty of... | https://mathoverflow.net/users/75536 | Serre's surjective theorem importance | There is an obvious application to the inverse Galois problem. And, some other instances of this problem have been extensively studied along the same lines (that is by proving surjectivity of some Galois reps attached to cuspidal modular forms for instance).
| 3 | https://mathoverflow.net/users/14967 | 218625 | 102,656 |
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