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https://mathoverflow.net/questions/79674 | 5 | Dear all,
I am thinking about a problem as follows:
Suppose a simply-connected 2-dimensional manifold has an $S^1$ boundary, is it homeomorphic to the open disk $D^2$? In fact, I would like to understand the general higher-dimensional case, i.e., how to decide the homeomorphism type of a (connected) manifold $M$ via ... | https://mathoverflow.net/users/9915 | Decide a manifold via its boundary | Yes, assuming the manifold compact. One way to see this is this: if we glue a disk along the boundary, we get a manifold which is simply-connected manifold by Seifert-van Kampen, and closed, hence a sphere by the classification theorem.
The same argument plus the Whitehead theorem and the topological Poincare conject... | 6 | https://mathoverflow.net/users/2349 | 79675 | 47,909 |
https://mathoverflow.net/questions/79534 | 16 | Take $n$ points uniformly in $[0,1] \times [0,1]$. Then pick uniformly $X\_0$ one of these points as your starting point. Then let $X\_1$ be the nearest neighbor of $X\_0$, let $X\_2$ be the nearest neighbor (not yet visited) of $X\_1$ and so on. What can be said of the asymptotic of $X\_n-X\_{n-1}$ the length of the l... | https://mathoverflow.net/users/18433 | Length of the last edge when visiting points by nearest neighbor order | This is closely related to a nice [open problem of David Aldous](http://www.stat.berkeley.edu/~aldous/Research/OP/greedy_tour.html), from the list of open problems on his web site, some version of which in fact has quite a long history in the combinatorial optimization community. At the above link Aldous has references... | 6 | https://mathoverflow.net/users/3401 | 79687 | 47,916 |
https://mathoverflow.net/questions/78560 | 2 | If you consider $f=\frac{P}{Q}$ the quotient of two polynomial function (i.e. $P,Q\in \mathbb{C} [z]$) then $\frac{f'}{1+|f|^2}$ decrease like $\frac{1}{z}$. My question is, is the converse true? is an meromorphic function(define on the whole plane) which satisfies
$$\frac{f'}{1+|f|^2}=O\left(\frac{1}{z}\right)$$
a... | https://mathoverflow.net/users/18642 | holomorphic function with special decreasing property | Some one give an almost complete answer on mathstackechange,
<https://math.stackexchange.com/questions/73651/holomorphic-function-with-special-decreasing-property>
the discussion should be continued there.
thanks all.
| 0 | https://mathoverflow.net/users/18642 | 79692 | 47,919 |
https://mathoverflow.net/questions/79681 | 6 | My chemist roommate asked me the following question. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real-valued function and $F$ its Fourier transform. Suppose we know the modulus function $|F| : \mathbb{R} \rightarrow \mathbb{R}$. What can we deduce about $f$, can we determine it completely?
Feel free to assume an... | https://mathoverflow.net/users/18540 | Fourier transform of a real-valued function. | To try to determine a function from the absolute value of its Fourier transform is actually the famous "hidden phase problem". In X-ray crystollography one measures the absolute value of the Fourier transform of a function that describes where the atoms in the molecule are located. However, using clever tricks and some... | 12 | https://mathoverflow.net/users/16546 | 79694 | 47,920 |
https://mathoverflow.net/questions/79693 | 13 | This is probably totally obvious but I have no clue how this is done: Say you have an endomorphism $f:X \rightarrow X$ of schemes. Why (if true, perhaps some additional assumptions are necessary!) do you get for a Zariski/étale/l-adic sheaf $\mathcal{F}$ on $X$ an induced endomorphism on the corresponding cohomology? H... | https://mathoverflow.net/users/18935 | Why does a group action on a scheme induce a group action on cohomology? | If a (say constant) group $G$ acts on a scheme $X$, you may want to consider the notion of a $G$-sheaf : a sheaf $\mathcal F$ endowed with isomorphisms $\lambda\_g: g^\* \mathcal F\simeq \mathcal F$, for $g\in G$ satisfying the usual cocycle conditions. Then by functoriality of cohomology for $g:X\to X$ you get an isom... | 17 | https://mathoverflow.net/users/11682 | 79697 | 47,922 |
https://mathoverflow.net/questions/79704 | 4 | I attended a course on stochastic processes a few years ago. During the course the lecturer mentioned that there is a mathematical proof (with some assumptions, naturally) of non-existence of a fair coin. Now I can't recall the details and can't locate the paper.
Is there such a proof?
I vaguely remember that the i... | https://mathoverflow.net/users/nan | Is there a fair coin? | This well-known paper seems to imply that the shape of the coin doesn't really matter:
<http://comptop.stanford.edu/u/preprints/heads.pdf>
From the lit review:
>
> In light of all the variations, it is
> natural to ask if inhomogeneity in the
> mass distribution of the coin can
> change the outcome. [Lindley, ... | 12 | https://mathoverflow.net/users/1256 | 79707 | 47,927 |
https://mathoverflow.net/questions/79708 | 7 | Can anybody explain me if the category of (associative) rings is co-well-powered? (This is the MacLane definition, in Russian literature this is called "locally small from the right side".) I mean, it is well-powered, of course, since for any ring A one can easily find a skeleton in the category Mono(A) (of all monomor... | https://mathoverflow.net/users/18943 | Is the category of rings co-well-powered? | Yes, rings as any algebraic theory make a locally presentable category, and this is well- copowred (Adámek and Rosicky, Locally Presentable and Accessible Categories Cambridge University Press, Cambridge, (1994))
About topological case, in the MAria Clementino article "Categorical and topological aspects
of semi-abe... | 10 | https://mathoverflow.net/users/6262 | 79725 | 47,934 |
https://mathoverflow.net/questions/79719 | 1 | I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.
In particular, I'm trying to interpret statements of the form
$$
\|\Sigma\_{n(p)} - \Sigma \| = O\_P\left( \frac{\log p}{n(p)}\right).
$$
As far as i can te... | https://mathoverflow.net/users/2586 | proofs of stochastic boundedness | The problem is that there are no universally agreed upon notations. Maybe [this](http://arxiv.org/abs/1108.3924) will help.
| 1 | https://mathoverflow.net/users/1061 | 79738 | 47,941 |
https://mathoverflow.net/questions/79746 | 1 | If you take a conic through 5 rational points on a quartic curve, then will at least one of the remaining 3 points also be rational ?
| https://mathoverflow.net/users/3537 | rational points on degree 4 curve | No. Let the conic be the union of lines L\_1 and L\_2, one of which intersects the quartic Q in four rational points. Now the intersection of L\_2 with Q is parametrized by the roots of a quartic equation which can be whatever it wants; in particular it can have one root rational and the rest not.
| 6 | https://mathoverflow.net/users/431 | 79750 | 47,947 |
https://mathoverflow.net/questions/79729 | 5 | If a plane curve of degree n intersects an elliptic curve in 3n points, then do those points always sum to zero when added using the group law on the points of an elliptic curve ?
| https://mathoverflow.net/users/3537 | Elliptic curve group law, Sum of intersection points | I assume that elliptic curve means a cubic curve $E$ with some base point $O$.
Then the answer to your question is yes if and only if the base point O for the group law is a flex point of $E$. (In many texts it is assumed that the base point of $E$ is a flex line, but to define the group law this is completely unnece... | 8 | https://mathoverflow.net/users/8621 | 79751 | 47,948 |
https://mathoverflow.net/questions/79764 | 4 | I have been trying to learn some deformation theory, and came across the following in a paper:
The first order deformations of a morphism of smooth curves $f:X\rightarrow Y$ is in bijection with $H^0(X,f^\*(\mathcal{T}\_Y))$.
I would like to understand a proof of this. I understand some simple facts, like $H^1(X,\m... | https://mathoverflow.net/users/3261 | Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf | There is actually the following general result.
Let us consider a morphism of algebraic schemes $f \colon X \to Y$, where $X$ is reduced and projective and $Y$ smooth. Then the first order deformations of $f$ *leaving the domain and the target fixed* are parametrized by $H^0(X, f^\* T\_Y)$.
For a proof (that does n... | 6 | https://mathoverflow.net/users/7460 | 79767 | 47,957 |
https://mathoverflow.net/questions/79761 | 8 | According to [this page](http://math.columbia.edu/~dejong/wordpress/?p=623) and thence [linked text](http://www.numdam.org/item?id=SAC_1967-1968__2__A2_0), if $e : R \to S$ is an epimorphism of rings, then the cardinality of $S$ cannot exceed the cardinality of $R$. This is a non-trivial observation because epimorphism... | https://mathoverflow.net/users/1176 | Why is the cardinality of the codomain of a ring epimorphism at most the cardinality of the domain? | An explanation of a layman to a layman.
Let $T={\rm Im}\ e$. Then embedding $T\to S$ is again an epimorphism. With respect to $T$ the ring $S$ behaves like the ring (not necessarily the field!) of fractions (compare $\mathbb{Z}$ and $\mathbb{Q}$), so it has the same cardinality. It is enough? :-) I think it is possible... | 2 | https://mathoverflow.net/users/18814 | 79775 | 47,964 |
https://mathoverflow.net/questions/79758 | 7 | Any ideas or references on how to approach this problem? Every element in a set has a parameter $p\_i \geq 0$ and $c\_i \geq 0$. The objective is to find a subset which maximizes
$\prod\_{i \in S} p\_i + \sum\_{i \in S} c\_i$
I tried to prove complexity results but was unsuccessful. My guess is that it is NP-Hard, ... | https://mathoverflow.net/users/18143 | Combinatorial Problem | It is actually an easy one. Suppose that you were allowed to take any part of each item adding $tc\_i$ and multiplying by $p\_i^t$ for any $t\in[0,1]$. First, you should take everything that has $p\_i\ge 1$ (because it never hurts). Then you have to maximize $C\prod\_j e^{-b\_j t\_j}+\sum\_j t\_j c\_j$ where $b\_j=-\lo... | 14 | https://mathoverflow.net/users/1131 | 79779 | 47,965 |
https://mathoverflow.net/questions/76294 | 1 | Hi,
Here are a couple of questions:
1. Is there a way to classify all homomorphisms between two finite posets?
2. Same question as (1) but for infinite, locally finite countable connected posets.
These questions are relevant to causal set quantization.
Thank you
PS: I mention a partial result concerning AUTOMOR... | https://mathoverflow.net/users/nan | Causal sets quantization | I think you should study rather endomorphisms of posets than homomorphisms. Then you obtain an algebraic structure (the semigroup of endomorphisms) and in addition a lot of articles in this problem. See, e.g.,
P. M. Higgins, J. D. Mitchell, M. Morayne and N. Ruškuc, Rank Properties of Endomorphisms of Infinite Partial... | -1 | https://mathoverflow.net/users/18814 | 79787 | 47,971 |
https://mathoverflow.net/questions/79785 | 2 | Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$. Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$. Call a set a ball-intersection if it is an intersection of closed balls with centers in $X$.
Does there exist a decreasing sequence of nonempty ball-inter... | https://mathoverflow.net/users/10583 | Geometry of the Hilbert sphere | I think yes. Your balls are of the form $X\cap S(x,r)$ where $x\in X$ and $S(s,r)$ is the slice
$\{y: \|y\|\le 1 \ \text{and} \ \langle x,y\rangle \ge r\}$ of the unit ball. Note that if $y$ is in a slice, so is $y/\|y\|$. The slices have non empty intersection because they are weakly compact.
EDIT: This argument loo... | 2 | https://mathoverflow.net/users/2554 | 79789 | 47,973 |
https://mathoverflow.net/questions/79515 | 4 | Let $G=KH$ be a frobenius group with non-abelian kernel $K$,
$|H|=r-1$,
$|K|=r^2$,
$r=2^m$ for some odd integer $m$,
$Z(K)=K'=\Phi(K)$, the Frattini subgroup of $K$,
$[K:K']=|K'|=r$.
Let both $K/K'$ and $K'$ be elementary abelian 2-groups.
My questions are:
1) In my special case, is it correct that <{$k... | https://mathoverflow.net/users/12826 | Specific question about the first omega subgroup of the non-abelian kernel of a frobenius group | Yes, this is true. Suppose that $\Omega\_1(K)=K$, and let $t\in K$ be such that $t^2=1$, but $t\notin Z(K)$. Then $M=\langle t,Z(K)\rangle$ is an elementary abelian subgroup. But $K=\cup\_{h\in H}\ M^h$, because $H$ acts transitively on $K/\Phi(K)$, and thus every element of $K$ has order $2$, which is absurd.
Note t... | 2 | https://mathoverflow.net/users/1446 | 79794 | 47,975 |
https://mathoverflow.net/questions/79793 | 1 | Let $\mathcal{D} \approx \mathbb{P}^{\delta\_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta\_d = \frac{d(d+3)}{2}$. Note that we have two tautological line bundles
$$ \gamma\_{\mathcal{D}} \rightarrow \mathcal{D}, \qquad \gamma\_{\mat... | https://mathoverflow.net/users/4463 | Does the diffeomorphism group preserving a particular section act transitively? | I am not sure I understand the meaning of the equation $\psi\circ g=\psi$ as I don't see how to compare a section at two different points without fixing an isomorphism between the line-bundle and its pull-back under $g$. Anyway, in any possible interpretation, I don't think the group preserving $\psi$ acts transitively... | 4 | https://mathoverflow.net/users/605 | 79796 | 47,977 |
https://mathoverflow.net/questions/79799 | 6 | The following assertion appears in a paper I am reading, and I can't seem to verify it.
Let $\text{Gr}\_{n,m}$ denote the set of pairs $(V,W)$ where $V$ and $W$ are as follows.
1. $V$ is an $n$-dimensional subspace of $\mathbb{C}^{\infty}$.
2. $W$ is an $m$-dimensional subspace of $\mathbb{C}^{\infty}$.
3. $V$ and ... | https://mathoverflow.net/users/18954 | Homotopy equivalence between the Grassmannian Gr_{n,m} and Gr_n \times Gr_m. | The forgetful map $Gr\_{n,m} \to Gr\_n$ that drops $W$ is a fiber bundle (exercise), and the map $Gr\_{n,m} \to Gr\_n \times Gr\_m$ is a map of fiber bundles. It's an equivalence on the (connected) base space, so it suffices to check that the map of fibers is an equivalence.
The fibers over $V$ are, respectively: $m$... | 10 | https://mathoverflow.net/users/360 | 79801 | 47,979 |
https://mathoverflow.net/questions/68630 | 2 | Here is the question:
if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH\_n(X) \to \prod\_{x \in X^{(0)}} KH\_n(k(x))$ injective?
If $X$ is smooth, this is known for the Zariski sheaf associated to $KH\_n$. I am wondering if anyone knows off the top of their h... | https://mathoverflow.net/users/12914 | Gersten for homotopy invariant K-theory of non-singular varieties. | Your question is an interesting one. But smoothness is essential to Quillen's proof of Gersten's conjecture (in his paper, Higher Algebraic K-theory : I), which, I'm assuming, you intend to use to deduce the consequence you state.
| 2 | https://mathoverflow.net/users/15247 | 79805 | 47,982 |
https://mathoverflow.net/questions/79809 | 2 | The Iwasawa decomposition and Cartan decomposition for $GL(n)$ is available for local fields. This can be proven for totally disconnected fields and archimedian fields seperatly by hand.
Here is a question, I am asking out of curiosity:
Is there a proof, which does not use the exact structure of the maximal compact... | https://mathoverflow.net/users/10400 | Iwasawa decomposition and Cartan decomposition | You have a proof which uses the fact that maximal compact open subgroups correspond to vertices in the building of ${\rm GL}(n)$. For instance for the Cartan decomposition you want to classify the orbits of $K={\rm GL}(n,{\mathfrak o}\_F)$ (${\mathfrak o}\_F$ denotes the ring of integers of your p-adic field $F$) in th... | 3 | https://mathoverflow.net/users/4767 | 79816 | 47,986 |
https://mathoverflow.net/questions/79817 | 2 | If we consider metric spaces to be categories enriched over $\mathbb R\_{\geq 0}$, the object corresponding to presheaves should be lipschitz-continuous functions $\operatorname{Lip^ 1}(M, \mathbb R\_{\geq 0})$. Now there should be an obvious metric on this set; making the Yoneda map $$x\mapsto \operatorname d(-,x)$$
a... | https://mathoverflow.net/users/1261 | Reference Request(Enriched Categories): Metric on Lipschitz Continuous Functions | It is the usual sup metric. See section 2 of Lawvere's original [article](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).
| 3 | https://mathoverflow.net/users/4262 | 79823 | 47,988 |
https://mathoverflow.net/questions/79807 | 9 | At [this nLab page](https://ncatlab.org/nlab/show/COSHEP) we have the line
>
> In contrast, any topos that violates countable choice, of which there are plenty, must also violate internal COSHEP.
>
>
>
It doesn't give an example, and neither does the [page on countable choice](https://ncatlab.org/nlab/show/cou... | https://mathoverflow.net/users/4177 | Example of a topos that violates countable choice | One sort of examples consists of the topoi of sets and functions obtained from models of ZF that violate countable choice. The original Cohen model is among these, and so are many others. Perhaps easier to understand are permutation models of ZFA (the variant of ZF that allows for atoms (= urelements)). The basic Fraen... | 13 | https://mathoverflow.net/users/6794 | 79828 | 47,989 |
https://mathoverflow.net/questions/79808 | 3 | Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions
$\psi = \psi\_1 + \psi\_2$
such that $\psi\_1$ and the Fourier transform of $\psi\_2$ are actually measurable functions of moderate growth. If so, under which additional conditions are the choices $\psi\_1$ and ... | https://mathoverflow.net/users/10400 | Decomposition of distributions | The [Dirac Comb](http://en.wikipedia.org/wiki/Dirac_comb), an infinite sum of delta functions, is an example of a tempered distribution that cannot be thusly decomposed (its Fourier transform is another Dirac Comb).
[Added:] There is a positive result in this direction that I (among others) only partly-remembered: A... | 4 | https://mathoverflow.net/users/6753 | 79839 | 47,998 |
https://mathoverflow.net/questions/79865 | 16 | A variety is called $\mathcal D$- affine if the global section functor induces an equivalence between quasi-coherent $\mathcal D$-modules and modules over $\Gamma(X,\mathcal D)$.
It is easy to see that affine varieties are $\cal D$-affine. More surprisingly, by an important theorem of Beilinson-Bernstein, (partial) fl... | https://mathoverflow.net/users/2837 | Examples for D-affine varieties? | As Alexander mentions above, it is a conjecture (and one that has stood for a while) that the OP has given a complete list of D-affine projective varieties.
My perspective on this is that $T^\*G/P$ (which differential operators quantize) are just not a good model for other cotangent bundles. For example:
* $T^\*G... | 21 | https://mathoverflow.net/users/66 | 79870 | 48,012 |
https://mathoverflow.net/questions/79872 | 2 | How does one compute Chern numbers of spherical rational homology classes
$$f: S ^{2k} \to BU.$$ These generate rational homology by Milnor-Moore theorem since BU is a connected H-space, and so $c\_k$ cannot kill such a class. It seems very likely that $\langle c\_k,[f] \rangle =1$ but what is the proof?
Let me add her... | https://mathoverflow.net/users/16877 | Chern numbers of primitive classes in BU | We have that if $f\colon S^{2k}\to BU$ is an actual map of topological spaces (it is a little bit unclear from your formulation if you assume this) then $\langle c\_k,[f]\rangle>$ is a multiple of $(k-1)!$ and all multiple are possible. See for instance Husemoller: Fibre bundles, Cor 18.9.8, GTM 20, Springer Verlag.
| 5 | https://mathoverflow.net/users/4008 | 79873 | 48,014 |
https://mathoverflow.net/questions/79829 | 4 | Assume I have a $n\times n$ positive semidefinite matrix $G$ of rank $p$ satisfying a set of $np - p(p-1)/2$ equations $v^T\_jGv\_j = 1$, $j = 1 \ldots np - p(p-1)/2$ for some given vectors $v\_j$. It is assumed these equations are linearly independent. Note here that the number of equations is exactly equal to the deg... | https://mathoverflow.net/users/18693 | Unique matrix satisfying a system of equations | In the case $n=3$,$p=2$, your 5 constraints for $v\_1 = (1,0,0)^T$, $v\_2 = (0,1,0)^T$, $v\_3 = (0,0,1)^T$, $v\_4 = (1,-2,0)^T$ and $v\_5 = (1,-1,1)^T$ have solution $G = \pmatrix{1 & 1 & t\cr
1 & 1 & t\cr t & t & 1\cr}$, which has rank 2 and is positive semidefinite if $-1 < t < 1$.
| 4 | https://mathoverflow.net/users/13650 | 79876 | 48,016 |
https://mathoverflow.net/questions/79811 | 5 | **The problem**:
I have a system of **N** linear equations, with **K** unknowns; and **K > N**.
Although the equations are over $\mathbb Z$, the unknowns can only take the values **0** or **1**.
Here's an example with **N**=11 equations and **K**=15 unknowns:
>
> $1 = x\_1 + x\_9$
>
> $2 = x\_{1} + x\_{2} +... | https://mathoverflow.net/users/18957 | How many 0, 1 solutions would this system of underdetermined linear equations have? | In complexity terms, no "efficient" (polynomial time) solution is likely.
However in practical terms you may be able to solve quite large problems of this nature, either by using integer linear programming software (I recommend Gurobi) or constraint satisfaction programming software.
For example, here is how you wo... | 5 | https://mathoverflow.net/users/1492 | 79880 | 48,018 |
https://mathoverflow.net/questions/79881 | 3 | That is wrong or right about this question and answer?
Question: Is there a cardinality which is greater than the continuum?
Answer: Yes and No. If there is a Universe where a given cardinal kappa is greater than the size of the continuum, then there is a Generic-Extension of this Universe where the size of the con... | https://mathoverflow.net/users/nan | A question about a question and answer. | In ordinary set theory, "Yes" is right and "No" is wrong. Even after you generically extend the universe to make the cardinal of the continuum bigger than a given $\kappa$, there are plenty of other cardinals that are even bigger than your new continuum. As M Turgeon says, to avoid cardinals larger than the continuum, ... | 10 | https://mathoverflow.net/users/6794 | 79882 | 48,019 |
https://mathoverflow.net/questions/79883 | 8 | Steven Weintraub's book {\em A Guide to Advanced Linear Algebra} includes the following remark:
"Of course, there is no algorithm for factoring polynomials, as we know from Galois theory."
I can't make sense of this. I feel confident that Galois theory doesn't speak to the question of algorithms, and confident that... | https://mathoverflow.net/users/10909 | Galois theory and algorithms | You are absolutely correct, this statement as stated does not make much sense. Over the integers (or any algebraic extension thereof), there are known algorithms for factoring multivariate polynomials. Any textbook on Computer Algebra will list some of them.
This has been an area of research with ups and downs, with ... | 11 | https://mathoverflow.net/users/3993 | 79885 | 48,020 |
https://mathoverflow.net/questions/79891 | 5 | I'd like to know if there exists a holomorphic rank 2 sub-bundle of $T\mathbb{P}^3$ which, when restricted to a given line is $\mathcal{O}(-a)\oplus \mathcal{O}(a)$, but is trivial when restricted to all other lines lying in a plane containing this line (i.e. this line is a jumping line of order $a$).
EDIT: From Ange... | https://mathoverflow.net/users/3709 | A vector bundle with a given jumping line | The only holomorphic subbudles of $T\mathbb P^3$ are the null-correlation bundles coming from symplectic forms in 4 variables (see for example <http://www.math.ubc.ca/~reichst/nesting.pdf>, Corollary 1.6). The first Chern class of a null-correlation bundle is non-zero, so the answer is negative.
| 7 | https://mathoverflow.net/users/4790 | 79896 | 48,027 |
https://mathoverflow.net/questions/76362 | 21 | **Short version:** One can define a version of the Lefschetz fixed point theorem using any homology or cohomology theory. All versions will be true on some topological spaces, since they agree on some topological spaces, but some might be true more generally than others. If two versions have the same generality, one mi... | https://mathoverflow.net/users/18060 | What is the best homology/cohomology theory for the Lefschetz fixed point theorem? | To answer your specific question about compact T3 spaces: First of all, every compact Hausdorff space (T2 space) is automatically a T4 space (a Hausdorff normal space). In the literature one usually says "compact Hausdorff space". A continuum is a compact, connected, metrizable Hausdorff space. There is a famous exampl... | 20 | https://mathoverflow.net/users/1450 | 79899 | 48,029 |
https://mathoverflow.net/questions/79850 | 13 | Does anyone have a recommendation for software which can efficiently calculate the Baker-Campbell-Hausdorff series in classical Lie algebras?
Right now, I have a problem which boils down to understanding Baker-Campbell-Hausdorff with respect to a basis in su(2), and this seems like the kind of thing Sage or Mathemat... | https://mathoverflow.net/users/9581 | Software for Computing Baker-Campbell-Hausdorff | There is a quite comprehensive package for Lie algebras in Maple. It is developed by Ian Anderson (from Utah State not Jethro Tull).
| 3 | https://mathoverflow.net/users/6818 | 79911 | 48,035 |
https://mathoverflow.net/questions/75522 | 8 |
>
> A vector space $V$ of dimension $n$ has an associated determinant line $Det(V)$.
>
> An element of $Det(V)$ is represented as a (formal limear combination) of expresstions of the form
> $v\_1 \wedge v\_2 \wedge \ldots \wedge v\_n$, subject to the usual multilinearity and antisymmetry relations.
>
>
>
I'... | https://mathoverflow.net/users/5690 | How can I write down a point in the Berezinian of a super vector space? | From page 61 of Deligne and Morgan's article *Notes on supersymmetry (following Joseph Bernstein)*:
>
> "A basis $\{e\_1,\ldots,e\_p,e\_{p+1},\ldots,e\_{p+q}\}$ of $L$ defines a one-element
> basis $[e\_1,\ldots,e\_p,e\_{p+1},\ldots,e\_{p+q}]$ of $Ber(L)$."
>
>
>
That's an answer to the question.
| 0 | https://mathoverflow.net/users/5690 | 79914 | 48,036 |
https://mathoverflow.net/questions/79913 | 3 | If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
| https://mathoverflow.net/users/18583 | Action on a compact group | If $G$ is a simple compact matrix Lie group of positive dimension, then there are continuously many orbits: elements with distinct eigenvalues are not conjugate and the outer automorphism group is finite.
If on the other hand $G=\mathbb{Z}\_p$ with $p$ a prime, there are countably many orbits.
On yet another hand, ... | 5 | https://mathoverflow.net/users/2349 | 79916 | 48,038 |
https://mathoverflow.net/questions/79869 | 5 | Let $(X,\mu,\mathcal{F})$ be a probability space. The paper *[Equiconvergence of Martingales](http://projecteuclid.org/euclid.aoms/1177693405)* by Edward Boylan introduced a pseudometric on sub-$\sigma$-fields (sub-$\sigma$-algebras) of $\mathcal{F}$ as follows:
$\rho(\mathcal{G},\mathcal{H})
:= \sup\_{A\in \mathcal... | https://mathoverflow.net/users/12978 | Is the Hausdorff metric on sub-$\sigma$-fields separable? | Take a sequence $A\_n$ of independent sets of measure $1/2$. Given two different subsets $B$ and $C$ of natural numbers, suppose WLOG that there is an $n$ in $B\sim C$. Now $\mu(A\_n\Delta A) = 1/2$ for all sets $A$ which are independent of $A\_n$, so the distance from the sigma algebra generated by $(A\_n)\_{n\in B}$ ... | 5 | https://mathoverflow.net/users/2554 | 79923 | 48,040 |
https://mathoverflow.net/questions/79918 | 0 | Let $A$ be an abelian scheme over a base scheme $S$ and $\omega$ a global section of the differential module $\Omega^1\_{A\times\_S A/S}$.
Suppose that $\omega$ is zero when restricted to $A\times S$ and $S\times A$, both times via the zero section and the identity.
Then why can one conclude that $\omega$ itself is... | https://mathoverflow.net/users/18183 | Differential forms on abelian schemes | It has nothing to do with abelian schemes. Just use the ``product rule'' of differentiation.
This is a natural isomorphism:
$$
p\_1^\*\Omega^1\_{A/S}\oplus p\_2^\*\Omega^1\_{A/S}\to \Omega^1\_{A\times\_S A/S}
$$
The zero section of $A$ affords an inverse to this natural map: if $s\_1 : A\to A\times\_S A$ is the inc... | 5 | https://mathoverflow.net/users/36285 | 79924 | 48,041 |
https://mathoverflow.net/questions/79818 | 0 | Hello everyone,
I'm currently working over a certain class of ODE of the form
$D\_N \; \phi(x) = \lambda^N \phi(x) \quad, \quad N>1$
where $D\_N = \delta\_N \delta\_{N-1} \cdots \delta\_1$ and $\delta\_k = (\frac{d}{dx}-A\_k(x)), \quad 1 \leq k \leq N$.
For various reasons my interest at the moment is focused o... | https://mathoverflow.net/users/18961 | On properties of Wronskians of ODE | I don't have a reference, a while back I learned in a paper by Percy Deift something that my help; it is easy to fill in the details with a bit of linear algebra. Write the equation as a system $\frac{d}{dx}F = M F \quad $ where $F$ is a function of $x$ taking values in $R^{n}$, and $M$ is a square matrix valued functi... | 2 | https://mathoverflow.net/users/15828 | 79933 | 48,045 |
https://mathoverflow.net/questions/79932 | 0 | Let $G$ be an undirected odd cycle. Let $f$ be a proper 3-coloring of $G$. If $w=v\_1v\_2...v\_k$ is a walk on $k$ vertices of $G$, let $f(w)=f(v\_1)f(v\_2)...f(v\_k)$. Let $W\_k=\{f(w)|w$ is a walk on $k$ vertices in $G\}$. Let $|W\_k|$ be the cardinality of $W\_k$.
Is it true that $\lim\_{k \to \infty} \frac{\log ... | https://mathoverflow.net/users/4250 | Counting walks on proper colorings of odd cycles | Certainly not. Take the coloring that is alternating black and white, except for a single red.
| 1 | https://mathoverflow.net/users/1061 | 79941 | 48,049 |
https://mathoverflow.net/questions/79577 | 8 | It is well known that the graded algebra $\mathcal{M}(1)$ of Modular forms for $\Gamma = PSL\_2(\mathbb{Z})$ is the polynomial algebra
$$
\mathcal{M}(1) = \mathbb{C}[E\_4, E\_6]
$$
where $E\_4$ and $E\_6$ are the Eisenstein series of weights 4 and 6, respectively. It is also true that, while $E\_2 = -\frac{1}{24} + \su... | https://mathoverflow.net/users/1703 | How do the rings of level $N$ quasi-modular forms related to the rings of modular forms? | Dear Simon:
Your assertion is right. Let $\Gamma$ be a subgroup of finite index in $SL\_2(Z)$. Then any quasi-modular form for $\Gamma$ can be written uniquely as a polynomial in $E\_2$ with coefficients which are modular forms for $\Gamma$. This is proved in a paper by Kaneko and Zagier, A generalized Jacobi theta fun... | 11 | https://mathoverflow.net/users/19003 | 79945 | 48,050 |
https://mathoverflow.net/questions/79943 | 12 | Consider the two types of Grassmannians Gr(2,7) and Gr(3,6) having their plucker embeddings in $\mathbb P^{20}$ and $\mathbb P^{19}$ respectivley. The first one is 10-dimensional and latter is 9-dimensional, so each having codimension 10. We can easily compute their defining equations and both of them are defined by 35... | https://mathoverflow.net/users/19001 | How Gr(2,7) and Gr(3,6) are related? | I don't know whether the ideals have the same kind of free resolutions, but $Gr(3,6)$ is definitely not a hyperplane section of $Gr(2,7)$. Otherwise, their $H^{\leq 8}$ would be the same by the Lefschetz hyperplane theorem. However, $H^6(Gr(3,6))$ is 3-dimensional and is spanned by $c\_1^3, c\_1c\_2$ and $c\_3$, and $H... | 11 | https://mathoverflow.net/users/2349 | 79948 | 48,052 |
https://mathoverflow.net/questions/79944 | 5 | Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
| https://mathoverflow.net/users/1937 | Elliptic curves on abelian surface | In general, if an abelian variety $A$ contains an abelian subvariety $B\subseteq A$, then $A$ contains another abelian subvariety $B'\subseteq A$ such that $A$ is isogenous to $B\times B'$. This is [Poincaré's reducibility theorem](http://books.google.com/books?id=MOW2gEP7HIkC&lpg=PP1&dq=birkenhake%2520lange&pg=PA125#v... | 19 | https://mathoverflow.net/users/10076 | 79951 | 48,054 |
https://mathoverflow.net/questions/79920 | 10 | What is the (currently known) consistency strength of global failure of the GCH?
I do not have access to the exact statement of the original Foreman-Woodin result. My searches seem to indicate that they used an assumption at the region of a supercompact, although I have seen comments stating that the result has been ... | https://mathoverflow.net/users/18995 | Failure of the GCH | The following quotations are taken from Matthew Foreman and W. Hugh Woodin, "The generalized continuum hypothesis can fail everywhere," *Ann. Math.* **133** (1991), 1–35.
>
> THEOREM. Let $\kappa$ be a supercompact cardinal with infinitely many inaccessible cardinals above $\kappa$. Then there is a partial ordering... | 12 | https://mathoverflow.net/users/3106 | 79955 | 48,058 |
https://mathoverflow.net/questions/79934 | 4 | Consider an ordinary Dirichlet series which is absolutely converge in some half plane Re s>c.
Question:Suppose it can be extended meromorphically to the whole complex plane with finite many poles.is it of finite order?If not,is it possible to construct a counterexample?
| https://mathoverflow.net/users/18286 | Analytic continuation of ordinary Dirichlet series | Well, a simple counter example is $$A(s)=\sum\_{n=1}^\infty a\_n n^{-s}= e^{\eta(s)},$$
where
$$\eta(s)=\sum\_{n=1}^\infty (-1)^{n-1} n^{-s}=(1-2^{1-s})\zeta(s).$$
This Dirichlet series is obviously meromorphic since it is in fact entire and it is also absolutely convergent on some half plane Re$(s)>c$. This entire fu... | 7 | https://mathoverflow.net/users/10811 | 79957 | 48,059 |
https://mathoverflow.net/questions/79827 | 4 | I have been reading "Combinatorial Rigidity" by Graver, Servatius and Servatius and I am interested in their chapter on rigidity in dimension $\geq$ 3. I have two questions.
1. What is the current status of the Henneberg conjecture (ie that every 2-extension of a 3-isostatic graph is isostatic?) The GSS book was pub... | https://mathoverflow.net/users/18964 | Isostatic graphs and the Henneberg conjecture | Question 2 can be addressed computationally by computing the ranks of generic rigidity matrices corresponding to the sequence of graphs.
I'll show below that the 6th 2-extension for one choice of sequence creates a non-isostatic set from an isostatic one (because a $K\_{6,6}$ is formed exactly then). I'm not sure ho... | 1 | https://mathoverflow.net/users/353 | 79963 | 48,063 |
https://mathoverflow.net/questions/79967 | 1 | If you derive a right exact functor $F$ you get a functor normally denoted by $RF$ on the derived category. Similarly, if you start with a left exact functor $G$ you get a functor normally denoted by $LG$. These are simply two triangulated functors on a triangulated categories. Suppose they are both defined in the same... | https://mathoverflow.net/users/36285 | derived functors and triangulated categories | I am not sure what your goal is with this question, but I think there is an inherent problem with your set up. $RF$ stands for a functor *derived from* $F$ and not just a functor on the derived category. If you consider the derived category only as a triangulated category via a forgetful functor then you are also forge... | 9 | https://mathoverflow.net/users/10076 | 79968 | 48,065 |
https://mathoverflow.net/questions/79960 | -4 | You can write the n roots of an n degree polynomial in terms of its n coefficients, i.e., "Vieta's" formulas.
You can solve this system of nonlinear equations using Newton's method and the Jacobian.
What I am missing is which part of this procedure violates n>5 unsolvable algebraically --aren't all the matrix ope... | https://mathoverflow.net/users/13403 | what part of using vieta's formulas violates quintic non-solvability? | The proper notion is "unsolvability with respect to a certain set of operations"; in the case of Galois-Abel's result regarding the quintic equation, this means that there will be no nice algebraic formula using just nth-roots, addition, etc. (Use Some encyclopedia for the proper set.) There are formulas for solving th... | 8 | https://mathoverflow.net/users/3206 | 79969 | 48,066 |
https://mathoverflow.net/questions/79959 | 32 | Let $k$ be a field, and $A$ a $k$-domain, so that the fraction field of $A$ has transcendence degree $n$ over $k$.
If $A$ is finitely-generated over $k$, then $A$ has Krull dimension $n$ (Theorem A in Eisenbud).
However, if $A$ is infinitely-generated, then it is possible for the dimension of $A$ to be less than th... | https://mathoverflow.net/users/750 | Krull dimension less or equal than transcendence degree? | It looks to me like the answer is **yes**.
Fix any strictly increasing chain of primes $P\_0 \subsetneq P\_1 \subsetneq \cdots \subsetneq P\_m$ in $A$ of length $m$; we'll prove that $m \leq n$. Choose elements $x\_i \in P\_i \setminus P\_{i-1}$ for $i = 1,\dots,m$.
Let $B \subseteq A$ be the $k$-subalgebra genera... | 48 | https://mathoverflow.net/users/778 | 79974 | 48,069 |
https://mathoverflow.net/questions/79956 | 11 | An affine scheme $X = Spec(A)$ is said to be smooth if for any closed embedding
$X\subset\mathbf A^n$, of ideal $I$, it is true that, locally on $x\in X$, the ideal $I$
can be generated by a sequence $f\_{r+1},\dots,f\_n$ such that their Jacobian has maximal rank.
My question is:
* Will the Jacobian of ANY set of $... | https://mathoverflow.net/users/36285 | Jacobian criterion for smoothness of schemes | Yes, the rank of the Jacobian matrix doesn't depend on the set of generators of $I$. The Jacobian matrix at $x$ represents the subspace generated by the differentials at $x$ of all $f\in I$.
Note that the rank of the Jacobian matrix at $x$ is computed in the fiber where $x$ lives, it has nothing to do with the base ... | 12 | https://mathoverflow.net/users/3485 | 79977 | 48,072 |
https://mathoverflow.net/questions/69589 | 7 | I have redone this question:
On $\mathbb R^n$ the Carleson Operator if defined by
$$Cf(x) = \sup\_{R>0} \left \vert \int\_{B\_R(0)} e^{2\pi i x\cdot \xi} \widehat{f}(\xi) d \xi \right \vert. $$ (In the previous version I had an incorrect version of this written down which lead to some stupid conclusions).
* For $n... | https://mathoverflow.net/users/1467 | Carleson's Theorem (on the Adeles and other exotic groups) | There is a p-adic analogue of Carleson's theorem. This was worked out by Hunt and Taibleson and you can find an exposition in Taibleson's book "Fourier analysis on local fields". (NB Taibleson did a lot of work extending Euclidean harmonic analysis results such as Carleson's theorem and boundedness of singular integral... | 5 | https://mathoverflow.net/users/7361 | 79978 | 48,073 |
https://mathoverflow.net/questions/79889 | 3 | To my knowledge, the strongest modularity lifting theorem known to date can be found in the preprint **Potential automorphy and change of weight** by Thomas Barnet-Lamb, David Geraghty, Toby Gee and Richard Taylor (accessible through their webpages).
A key idea in this work seems to be the introduction of a new loca... | https://mathoverflow.net/users/11928 | Potential diagonalizability: motivation? | Graff's answer is (naturally) accurate. Historically I think it went something like this:
* observation that the fact that "automorphy of a point on a component of a universal deformation ring implies automorphy of the whole component" (i.e. the general TWK method) could profitably be combined with the Harris tensor ... | 6 | https://mathoverflow.net/users/1125 | 79981 | 48,075 |
https://mathoverflow.net/questions/79662 | 1 | Let $\langle M\_i:i<\theta\rangle$ be an increasing chain of Banach spaces, where each $M\_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M\_i$ is $\mu$). Let $B\_i\subset M\_i$ be a dense subset of $M\_i$ of cardinality $B\_i$. Notice that $\bigcup\_{i<\theta}B\_i$ is a dense subset... | https://mathoverflow.net/users/18925 | A question about density character of Banach spaces. | If $X$ is any metric space and $Y$ is any subspace of $X$ then $dc(Y) \leq dc(X)$.
| 3 | https://mathoverflow.net/users/17836 | 79982 | 48,076 |
https://mathoverflow.net/questions/79929 | 2 | Let me first state the definitions :
A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ; $ A closed curve / loop $c$ is called primitive if in the fundamental group $\pi\_1(X,c(1)),$ the homotopy class $[c]$... | https://mathoverflow.net/users/6953 | How to rigorously prove that simple closed curves on a surface are primitive closed curves ? | Suppose that $c = \gamma^n$ in $\pi\_1(X)$. Note that, as $\pi\_1(X)$ is torsion free and $c$ is assumed to be non-trivial, the element $\gamma$ generates an infinite cyclic subgroup $\langle \gamma \rangle < \pi\_1(X)$. Let $A = X^\gamma$ be the cover of $X$ corresponding to the subgroup $\langle \gamma \rangle$. So $... | 9 | https://mathoverflow.net/users/1650 | 79985 | 48,078 |
https://mathoverflow.net/questions/79927 | 4 | By [Robin's theorem](http://mathworld.wolfram.com/RobinsTheorem.html)
$$G(n)=\frac{\sigma(n)}{n \log \log n}$$
is bounded by $e^\gamma \approx 1.78107241799$ for $n>5040$ assuming Riemann hypothesis .
For $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ appears generally increasing as $k$ increases reaching $\approx 1.781... | https://mathoverflow.net/users/12481 | Which $n$ maximize $G(n)=\frac{\sigma(n)}{n \log \log n}$? | First, from a more detailed theorem of Robin we have an unconditional result (1984) that says that your ratio of interest is, for $n \geq 13,$ smaller than
$$ e^\gamma + \frac{0.64821364942...}{(\log \log n)^2},$$ with the constant in the numerator giving equality for $n=12.$
from which it follows that your supremum i... | 9 | https://mathoverflow.net/users/3324 | 79987 | 48,080 |
https://mathoverflow.net/questions/79984 | 7 | I want to understand Prop 6 in the paper "[Convergence of the Q-curvarture flow on $S^4$](http://dx.doi.org/10.1016/j.aim.2005.07.002)" by Simon Brendle. I understand that for every $p\in B^4$, where $B^4=\{x\in\mathbb{R}^5: |x|\leq 1\}$,
$$\phi(x)=p+\frac{1-|p|^2}{1+2\langle p,x\rangle+|p|^2}(x+p)
$$defines a conform... | https://mathoverflow.net/users/14579 | conformal diffeomorphism of sphere | Conformal diffeomorphisms of $S^n$ correspond to hyperbolic isometries of hyperbolic space $\mathbb H^{n+1}$ -- the idea is to think of $S^n$ as the visual sphere for hyperbolic space, all conformal diffeos extend uniquely to a hyperbolic isometry.
For (ii), no. Hyperbolic isometries have various forms. Your $\phi$ ... | 4 | https://mathoverflow.net/users/1465 | 79990 | 48,082 |
https://mathoverflow.net/questions/80002 | 47 | Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right?
In other words, do there exist finite groups $A$, $B$ and homomorphisms $f: A \rightarrow A \times B$, $g: A \times B \rightarrow B$ such that $1 \rightarro... | https://mathoverflow.net/users/19012 | Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups? | This is [true](https://doi.org/10.1515/JGT.2006.020) (1). It was extended to finitely generated profinite groups [here](https://doi.org/10.1515/JGT.2006.021) (2). Surprisingly, it is [also true](https://doi.org/10.1215/kjm/1250524308) in the category of finitely generated modules over a Noetherian commutative ring (3).... | 53 | https://mathoverflow.net/users/2083 | 80003 | 48,085 |
https://mathoverflow.net/questions/79999 | 13 | Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace 0,\ldots,n-1\rbrace \times \lbrace 0,\ldots,n-1\rbrace$ so as to maximize the sum, over all $p$ in $S$, of the dot pro... | https://mathoverflow.net/users/431 | Complexity of a weirdo two-dimensional sorting problem | To elaborate on the comments of Will Sawin and fedja: The question isn't a sorting problem, but it is a matching problem. If $S$ is your arbitrary set and $G = [n]^2$ is your grid, then you are marrying elements of $S$ to elements in $G$, where the happiness of each marriage is your dot product $p \cdot f(p)$. Any happ... | 19 | https://mathoverflow.net/users/1450 | 80004 | 48,086 |
https://mathoverflow.net/questions/80013 | 9 | Let $\mathcal{A}$ be a small category with some ( maybe no) colimits. What I would like to be able to do is add the rest of the colimits in a universal way. The Yoneda lemma will not work, since this simply adds all colimits formally. That is to say that you have new colimits that are different than the old. We do have... | https://mathoverflow.net/users/16801 | Given a small category with some colimits, can the rest of the colimits be added? | Yes.
More generally, let $A$ be a small category, and $D$ some set of "distinguished" colimits in $A$ — for example, you could take the set of all colimits that exist. A *sheaf* on $(A,D)$ is a functor $F: A^{\mathrm{op}} \to \mathrm{Set}$ such that for every colimit diagram $d\in D$, $Fd$ is a limit diagram in $\mat... | 10 | https://mathoverflow.net/users/78 | 80017 | 48,090 |
https://mathoverflow.net/questions/79723 | 0 | Let $\mathcal{D} \approx \mathbb{P}^{\delta\_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta\_d = \frac{d(d+3)}{2}$. Note that we have two tautological line bundles
$$ \gamma\_{\mathcal{D}} \rightarrow \mathcal{D}, \qquad \gamma\_{\mat... | https://mathoverflow.net/users/4463 | What is the simplest way to show that a section of a vector bundle is transverse to the zero set | It sounds like a multijet transversality theorem (see for example M. Golubitsky, V. Guillemin, Stable mappings and their singularities) in context of algebraic geometry. So, the answer is true --- a proof of this theorem uses only polynoms for perturbations which achieves general position.
| 1 | https://mathoverflow.net/users/4298 | 80018 | 48,091 |
https://mathoverflow.net/questions/79942 | 1 | I apologize if this is a simple question, but does the Grassmannian of lines in $\mathbb P\_k^3$, $\mathbb G(1,3)$, embed into $\mathbb P\_k^5$ when $k$ an algebraically closed field of characteristic $2$?
| https://mathoverflow.net/users/19000 | Plucker embedding in char 2 | The most general form of the Plücker embedding I know of is the following, which you can find in EGA I, 9.8: Let $S$ be an arbitrary base scheme and $\mathcal{F}$ be a quasi-coherent sheaf on $S$. Then there is a canonical morphism of $S$-schemes $\mathrm{Grass}\_n(\mathcal{F}) \to \mathbb{P}(\Lambda^n(\mathcal{F}))$. ... | 1 | https://mathoverflow.net/users/2841 | 80028 | 48,094 |
https://mathoverflow.net/questions/80025 | 8 | The Duflo map is the map S(g) -> U(g), which known to satisfy the following properties:
1) identity on g
2) isomorphism of g-modules (and in particular vector spaces)
3) restricted to Poisson center on S(g) it is ISOMORPHISM of commutative algebras
S(g)^g to ZU(g) (the center of U(g)).
(This is highly non-trivial... | https://mathoverflow.net/users/10446 | Is the Duflo map for Lie algs. unique ? | Choose a map $\varphi$ satisfying these properties and make the difference $\psi=\varphi^{-1}\varphi\_D$ with the Duflo map. Then $\psi$ is an automorphism of the $\mathfrak g$-module $S(\mathfrak g)$ which is the identity on $\mathfrak g$ and is multiplicative on invariants.
If you want this map to be universal (na... | 7 | https://mathoverflow.net/users/7031 | 80045 | 48,104 |
https://mathoverflow.net/questions/80056 | 27 | I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the possible pros and cons of doing this.
Obviously, slides make it possible to produce and show clear graphs/pictures (whic... | https://mathoverflow.net/users/3635 | Using slides in math classroom | I think you already touched on the two main points: pretty pictures are so much better than anything done on a chalkboard is the pro, but you cannot decently unwind any argument on slides.
I've used them intensively, I do it a lot less now. (Here's a con you did forget about: they take a **lot** of time to prepare, ... | 26 | https://mathoverflow.net/users/8212 | 80058 | 48,109 |
https://mathoverflow.net/questions/80055 | 6 | While looking for a closed form of a expression I worked myself to a formula that resembles the Vandermonde convolution, but is summed over even binomial coefficients only.
$\sum\_{k=0}^n\sum\_{l=0}^n{{2k+2l}\choose{2l}}{{4n-2k-2l}\choose{2n-2l}}$
I'm at a loss as to what to do with it. I can re-write it in severa... | https://mathoverflow.net/users/19024 | Sums of binomials with even coefficients | We have
$$\sum\_l\binom{a+l}lx^l=\frac1{(1-x)^{a+1}},$$
hence the generating function for the even terms of the sequence is
$$\sum\_l\binom{a+2l}{2l}x^{2l}=\frac12\left(\frac1{(1-x)^{a+1}}+\frac1{(1+x)^{a+1}}\right).$$
Consequently,
\begin{multline\*}\sum\_l\binom{a+2l}{2l}\binom{b+2(n-l)}{2(n-l)}=\\\\
[x^{2n}]\frac14\... | 15 | https://mathoverflow.net/users/12705 | 80068 | 48,114 |
https://mathoverflow.net/questions/80061 | 2 | Let $X$ be a complex, projective, nonsingular variety. We also understand it as a Kähler Manifold. My question now is, when people say $c\_1(X) < 0$, what exactly do they mean? Let me elaborate. In [this paper](http://www.math.cuhk.edu.hk/~kwchan/MYIneq.pdf), it is said that Yau's inequality
$$ (-1)^n c\_1^n \le (-1)... | https://mathoverflow.net/users/9947 | Condition on the canonical divisor for Yau Inequality - effective or ample? | You should read $c\_1(X)<0$ as saying that the first Chern class of $T\_X$ is *negative*, or the line bundle $K\_X$ is *positive*, in the sense of curvature. But positive line bundles are ample line bundles. This fact is sometimes called the Kodaira Embedding Theorem. See for example p. 181 of Griffiths-Harris, Princip... | 4 | https://mathoverflow.net/users/7399 | 80070 | 48,116 |
https://mathoverflow.net/questions/80081 | 45 | I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:
>
> What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (or not spin) (and why)?
>
>
>
---
For comparison, I'd consider the cylinder and the mobius strip to be "go... | https://mathoverflow.net/users/1540 | What are "good" examples of spin manifolds? | There's the traditional obstruction-theoretic perspective. Orientability means the tangent bundle trivializes over a 1-skeleton. Dually you could think of that as saying the complement of a co-dimension $2$ subcomplex has a trivial tangent bundle.
So admitting a spin structure is the same, but it will be the tangent... | 38 | https://mathoverflow.net/users/1465 | 80090 | 48,128 |
https://mathoverflow.net/questions/80099 | 0 |
>
> Let $S\_{\kappa}$ denote the symmetric group on some set of cardinality $\kappa$. Does there exist a generating set $X \subset S\_{\kappa}$ such that $|X| < |S\_{\kappa}|$ ($\stackrel{?}{=} 2^{\kappa}$)?
>
>
>
More specifically, does there exist a countable set of generators for $S\_{\mathbb{N}}$? And if so... | https://mathoverflow.net/users/18573 | For the symmetric group on an infinite set, is there a generating set of strictly smaller cardinality? | It seems clear that the answer to the first and third questions is 'no'. Indeed, if a set of generators $X$ is of infinite cardinality $\alpha$, then the group so generated cannot have cardinality greater than $\alpha$, since it is a quotient of the free group generated by $X$, which in turn is a quotient of the free m... | 5 | https://mathoverflow.net/users/2926 | 80102 | 48,136 |
https://mathoverflow.net/questions/80075 | 15 | I have a good motivation to ask the question below, but since the post is
already a little long, and the problem looks rather natural and appealing
(well, to me, at least), I'd rather go straight to the point.
Let $n\ge 3$ be an integer. If $E$ denotes the standard basis of the
vector space ${\mathbb F}\_2^n$, then f... | https://mathoverflow.net/users/9924 | The hypercube: $|A {\stackrel2+} E| \ge |A|$? | OK, suppose that $n\ge 3$, let $A$ be a set of *even* vertices of cardinatily $2^n\mu\ge 2^{n-2}$ (so $\mu\ge \frac 14$), and write $B:=A{\stackrel2+}E$; that is, $B$ is the set of odd vertices with at least two neighbors in $A$. Assume that $|B|=2^n\xi$. Our aim is to show that $\xi\ge\mu$. Let us consider the action ... | 8 | https://mathoverflow.net/users/1131 | 80104 | 48,138 |
https://mathoverflow.net/questions/80125 | 21 | Prove/ Disprove: Let $n$ be a positive integer. Let $A$, $B$ be two $n \times n$ square matrices over the complex numbers. If $AB = BA$ and $\ker A = \ker A^2$ and $\ker B = \ker B^2$
then $\ker AB = \ker A + \ker B$.
(Recall that $\ker A$ is the set of all vectors $v$ such that $Av = 0$.)
Background: I am teachin... | https://mathoverflow.net/users/4048 | When is $\ker AB = \ker A + \ker B$? | Since $\ker A = \ker A^2$, the map $\bar{A} : V/\ker A \to V/\ker A$ is injective. Since $V/\ker A$ is finite dimensional, this map is surjective. So for any $x \in V$ we can find $y \in V$ and $z \in \ker A$ such that $x = Ay + z$.
Now suppose $ABx = 0$ and let $x = Ay + z$ as above. Then $0 = ABx = ABAy + ABz = A^2... | 28 | https://mathoverflow.net/users/6827 | 80129 | 48,147 |
https://mathoverflow.net/questions/78587 | 1 | Let $X\_1,\ldots X\_k$ be irreducible(may be singular) affine real algebraic hypersurfaces in $R^n$ with $x\_1,\ldots, x\_k$ connected components, respectively.
Let $G\_1,\ldots, G\_l$ be their intersections of dimension $n-2$, with $g\_1,\ldots, g\_l$ connected components, respectively.
How we can estimate an uppe... | https://mathoverflow.net/users/16044 | Number of connected components of complement to a reducible real algebraic hypersurface.[EDITED] | One possible upper bound could be found in a paper by Hugh E. Warren Lower Bounds for approximation by nonlinear manifolds//Transactions of the AMS. 1968. Vol.133 P.~167--178.
He gives the following bound:
Let $p\_1,\ldots, p\_m$ be real polynomials in $n$ variables, each of degree $d$ or less. Let $N(p\_i)$ be a s... | 2 | https://mathoverflow.net/users/16044 | 80136 | 48,151 |
https://mathoverflow.net/questions/80133 | 5 | I am looking for an article, most likely from the 90s, that generalized the bijection between partitions with odd and distinct parts by explaining how a bijection between the forbidden parts could be transformed into a bijection of the partitions.
(So, in the example above, a bijection can be formed using the fact th... | https://mathoverflow.net/users/14102 | Article about partitions with forbidden parts/multiplicities | Possible articles you might have seen could be
>
> M.V. Subbarao, [Partition theorems for Euler pairs](http://www.jstor.org/pss/2037963), Proc. Amer. Math. Soc. 28 (1971),
> no. 2, 330-336.
>
>
>
Here the author characterizes all triples $(A,B,r)$ so that the number of partitions with parts in $A$ is equal to... | 9 | https://mathoverflow.net/users/2384 | 80138 | 48,152 |
https://mathoverflow.net/questions/80135 | 1 | I'm reading Fulton's "Intersection theory", which i need for some applied needs.
And i have two questions on general definition of degree used in Fulton.
1)Let us we have a real algebraic variety defined by a set of equations
$f\_1=0, f\_2=0,\ldots ,f\_n=0$ of degrees $d\_1,\ldots, d\_n$ respectively.
Using a well-kn... | https://mathoverflow.net/users/16044 | Degree of a real algebraic variety and regular morphisms | I would encourage you to read **Algorithms in Real Algebraic Geometry**
by Saugata Basu, Richard Pollack, Marie-Françoise Roy, which contains all the state of the art results about effective results in real algebraic geometry. It is a free download from <http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.... | 4 | https://mathoverflow.net/users/8212 | 80140 | 48,153 |
https://mathoverflow.net/questions/78191 | 5 | In their great paper "Symplectic reflection algs. and Harish-Chandra hom."
<http://arxiv.org/abs/math/0011114>,
Etingof and Ginzburg write (page 9):
"In 1964, Harish-Chandra [HC] defined an algebra homomorphism $\Phi: D(\mathfrak{g})^{\mathfrak{g}} \rightarrow D(\mathfrak{h})^W$
that reduces to the restriction map: $... | https://mathoverflow.net/users/10446 | Harish-Chandra homom (Etingof-Ginzburg) for inv. dif. opers. on $\mathfrak{gl}_n$; images of higher order $\Delta$: $tr(D^3) = d_{ij}d_{jk}d_{ki}$ | I asked Pavel Etingof he answered that "naive answer is correct".
Means that higher order Laplacians are mapped to their naive restrictions.
This is explicitly stated in Proposition 4.5 page 27 in
<http://arxiv.org/abs/math/0606233>
P. Etingof "Lectures on Calogero-Moser"
===
Some steps of the proof - for qua... | 2 | https://mathoverflow.net/users/10446 | 80142 | 48,154 |
https://mathoverflow.net/questions/80151 | 1 | Let $X$ be a smooth variety over a field $\Bbbk$, and let $Y, Z \subset X$ be closed reduced subschemes of the same dimension, both of which are local complete intersections. Is $Y \cup Z$ necessarily a local complete intersection?
| https://mathoverflow.net/users/5094 | Is a union of local complete intersections, a local complete intersection? | The union of two planes in $\mathbb A^4$ which meet at a point is not Cohen--Macaulay, and so in particular not a local complete intersection.
More generally, any smooth subvariety of a smooth variety is a local complete intersection, so any non-Cohen--Macaulay subvariety whose components are smooth gives an example... | 8 | https://mathoverflow.net/users/2874 | 80153 | 48,159 |
https://mathoverflow.net/questions/80146 | 56 | Does anybody know if there exists a mathematical explanation of the Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the authors present quantum mechanics as an axiomatic system, so one could expect that there is a deduction from the axioms to ... | https://mathoverflow.net/users/18943 | Is the Mendeleev table explained in quantum mechanics? | I doubt any answer will be satisfactory. My opinion is that we are still very far from a mathematical justification. If we accept the mathematical foundations of quantum mechanics, and if we make the approximation that the nucleus of the atom is just one heavy thing with $N$ positive charges, then the motion of the $N$... | 21 | https://mathoverflow.net/users/8799 | 80155 | 48,161 |
https://mathoverflow.net/questions/80118 | 5 | If $X\_{i}$ are a bunch of iid random variables with mean 0 and finite second moments, we know that $\sum\_{i=1}^{n} \frac{X\_{i}}{\sqrt{n}}$ converges in law to a Gaussian. Furthermore, by the Berry-Esseen theorem, we have some bounds on the rates of this convergence. Similar results hold, even if $X\_{i}$ are only mi... | https://mathoverflow.net/users/19033 | Stable Law with Rates | Check out this paper:
Harmonic mean, random polynomials and stochastic matrices
Natalia L. Komarova, , Igor Rivin
There are some results of the sort you are asking about there (and since the authors were not then and are not now probabilists, the proofs start from essentially nothing).
| 0 | https://mathoverflow.net/users/11142 | 80158 | 48,163 |
https://mathoverflow.net/questions/80150 | 63 | I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible.
Let $\mathfrak{g}$ be a semisimple Lie algebra (say over $\mathbb{C}$) and $\mathfrak{h} \subset \mathfrak{g}$ a Cartan subalgebra. All the references I have seen whi... | https://mathoverflow.net/users/3544 | What is significant about the half-sum of positive roots? | I don't think there is a one-line answer to this question, since it depends a lot on the direction from which you approach semi-simple Lie theory. For one thing, it's probably best at first to emphasize just *integral* weights, among which the dominant ones parametrize irreducible finite dimensional representations. He... | 43 | https://mathoverflow.net/users/4231 | 80162 | 48,167 |
https://mathoverflow.net/questions/80169 | 5 | I am looking for a reference for a proof of the following:
Let $n$ and $a,b, \ldots ,z$ be non-negative integers with $a + b
+ \ldots + z = n$, and let $p$ be a prime. Write $n = n\_0 + n\_1 p + \ldots +
n\_m p^m$ in $p$-ary notation, similarly for $a, b, \ldots , z$.
Then, modulo $p$, the multinomial coefficient ${n... | https://mathoverflow.net/users/19048 | Reference needed for Lucas' Theorem for multinomial coefficients modulo a prime | You can prove this by induction on the maximum number of base $p$ digits, and to make the argument simpler it's better to formulate a mildly *stronger* theorem where the leading base $p$ "digit" is allowed to be nonnegative rather than be constrained between 0 and $p-1$: for $d \geq 0$, $t \geq 1$ and nonnegative integ... | 4 | https://mathoverflow.net/users/3272 | 80179 | 48,176 |
https://mathoverflow.net/questions/80176 | 6 | I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression
$
\sum\_{k=0}^{\infty} \frac{z^k}{k!k!} = F\_{1}[;1;z]
$
This is not very helpful though, and I 'd like to know if it is possible to get some asymptotic expansion for this function and if there exists some general approa... | https://mathoverflow.net/users/12418 | Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$ | This particular function can be expressed in terms of Bessel's I function as $I(0,2\sqrt{z})$, and from there an asymptotic expression (at $\infty$) is easily derived. It starts
$$\frac{e^{\frac{2}{\sqrt{\frac{1}{z}}}}\left(\frac{1}{z}\right)^{\left(\frac{1}{4}\right)}}{2\sqrt{\pi}} + O\left(e^{\frac{2}{\sqrt{\frac{1}{... | 7 | https://mathoverflow.net/users/3993 | 80185 | 48,178 |
https://mathoverflow.net/questions/80163 | 17 | Let $G$ be a finite group (maybe this will also work when $G$ is compact, or something, but to be safe we'll let it be finite). I imagine it's quite natural to ask: is the category of $G$-spectra equivalent to the category of module spectra over some ring spectrum, probably denoted by $SG$?
For definiteness, we can t... | https://mathoverflow.net/users/6936 | Are $G$-spectra the same as modules over a "group ring spectrum"? | Eric wrote a really nice response telling that your initial hope is incorrect and why. I'd just like to write some positive results that you can find.
Disclaimer: I understand little to nothing about the case of a compact Lie group.
Schwede and Shipley have a paper entitled "Stable model categories are categories o... | 17 | https://mathoverflow.net/users/360 | 80192 | 48,182 |
https://mathoverflow.net/questions/79830 | 16 | The Serre intersection formula, as an alternating sum of contributions from Tor-groups, is something that combines a lot of ingredients that I'm interested in, but I've never really felt that I have a "grip" on it. One of the reasons for this is that, despite making attempts on a couple of occasions, I never seem to ha... | https://mathoverflow.net/users/360 | Geometric examples of the Serre intersection formula | Consider a flat morphism $f:X\to Y$ of smooth connected varieties. For instance let $X=Y\times F$ with $X,Y,F$ all smooth. Further let $Z\subset X$ be a generically reduced subvariety such that $f|\_Z:Z\to Y$ is a finite morphism, which is **not** flat. For any $y\in Y$ let $X\_y\subset X$ denote the fiber of the origi... | 10 | https://mathoverflow.net/users/10076 | 80205 | 48,188 |
https://mathoverflow.net/questions/80194 | 3 | I asked this question on MSE, but didn't get enough information. If it is a violation of some norms, let me know, I'll delete it.
I'm having problem solving this difference equation. Initially I thought it should be quite easy to solve using generating functions (e.g. like in Migdal(2010), Woodbury(1949) or Gani(2006... | https://mathoverflow.net/users/12418 | Difference equation $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$ | Let
$$P\_m(x,z) = \prod\_{k=0}^{m-1} (p(x-k)+q(x-k)z)$$
and
$$\mathcal{A\_n}(x,z) = \sum\_{k=0}^{\infty} A(n,x-k) z^k.$$
Then unrolling the given recurrence $m$ times, we get that $A(n,x)$ equals the coefficient of $z^m$ in
$$P\_m(x,z)\cdot \mathcal{A}\_{n-m}(x,z).$$
In particular, for $A(n,x)$ equals the coefficien... | 3 | https://mathoverflow.net/users/7076 | 80206 | 48,189 |
https://mathoverflow.net/questions/80207 | 6 | hi,
does anybody know a good book on calabi yau manifolds (i am a beginner) ?
thanks in advance
lois
| https://mathoverflow.net/users/19053 | book on calabi yau manifolds | Depending on how much of a beginner you are, you could begin by reading Barth-Hulek-Peters-Van de Ven paying particular attention to the section on K3 surfaces (which are 2-(complex)-dimensional Calabi-Yaus):
<http://www.springer.com/mathematics/algebra/book/978-3-540-00832-3>
For an overview, you could try:
<htt... | 11 | https://mathoverflow.net/users/10839 | 80209 | 48,190 |
https://mathoverflow.net/questions/80199 | 3 | It is well known (and wouldn't be so-named unless it were) that:
If $\xi$, $\eta$ are $n$-fold extensions of $N$ by $M$ (modules over a ring $R$) which yield the same element of $\text{Ext}^n(M,N)$, then they are in fact equivalent.
I am trying to understand a certain proof of this fact (and indeed the proof, for m... | https://mathoverflow.net/users/19048 | Extensions which define the same element of $\text{Ext}^n(M,N)$ are in fact equivalent | Say the maps in your first displayed diagram are, left to right between the first two rows, $\psi, f\_1, f\_0$ and between the second two rows $\phi, g\_1, g\_0$. Let $\sigma \partial\_2 = \phi-\psi$ (I'm afraid my maps compose in the opposite direction to yours). Let the map $N \to Y\_1$ be $\iota\_Y$. You ask about t... | 2 | https://mathoverflow.net/users/6481 | 80210 | 48,191 |
https://mathoverflow.net/questions/79826 | 11 | ### Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X\_T$$^\*$, take etale cohomology, get a virtual character $R\_T^\theta$, maybe it's reducible, so you try to decompose it.
### Gelfand-Graev character
is awesome. You take a maximal unipotent ... | https://mathoverflow.net/users/2024 | Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety? | In the 35 years following the Deligne-Lusztig construction of generalized characters of finite groups of Lie type, a considerable amount of work by Lusztig and others has led to a reasonably detailed understanding of the irreducible characters. This is usually quite difficult to make explicit, however, since a lot of r... | 9 | https://mathoverflow.net/users/4231 | 80215 | 48,194 |
https://mathoverflow.net/questions/64883 | 9 | This question is not very precise; I hope it is suitable for the site.
I have come to a situation where I have to study rational points on an elliptic curve defined over $\mathbb{Q}$. I don't know much about the curve, let alone its equation. I already have one rational point, which sits on a bounded real connected c... | https://mathoverflow.net/users/828 | Elliptic curves with Mordell-Weil group Z/2Z over Q | Mazur's theorem ensures that there are exactly 15 possible cases for the torsion part of the Mordell-Weil group of an elliptic curve: the cyclic groups $\mathbb{Z}\_n$ (with $1\leq n\leq 10$ or $n=12$) and the groups $\mathbb{Z}\_2\times\mathbb{Z}\_n$ for $n=2,4,6,8$.
In his paper [Universal Bounds on The Torsion of ... | 14 | https://mathoverflow.net/users/4046 | 80216 | 48,195 |
https://mathoverflow.net/questions/80191 | 2 | Start with a category $C$. Form a monoid $M$ whose elements are lists of morphisms in the category $C$ subject to commuting diagrams in $C$. Is there a name for this construction or a better way to categorially understand this?
| https://mathoverflow.net/users/nan | The monoid of lists of morphisms in a category subject to commuting diagrams | Do you mean take the monoid with the following presentation? Take the arrows of C as generators and add the relations that f.g = fg if f and g are composable and that each identity of C be equivalent to 1? I would call this the universal monoid U(C) of C. It has the universal property that there is a functor $F:C\to U(... | 4 | https://mathoverflow.net/users/15934 | 80222 | 48,198 |
https://mathoverflow.net/questions/79777 | 39 | I was thinking about infinite exponential representation of real numbers (like $2=e^{e^{-e^{-e^{e^{-e^{e^{e^{-e^{-e^{-e^{-e^{-e^{e^{-e^{e^{e^{-e^{e^{\cdot^{\cdot^{\cdot}}}}}}}}}}}}}}}}}}}}}$.
The sequence of signs before exponents can be obtained by repeated application of $\ln|x|$ to $2$ and taking a sign of each resu... | https://mathoverflow.net/users/9550 | Infinite exponential representation of real numbers | First, even though I think this is a fun question,
it's not really research mathematics and I'm not sure it belongs on mathoverflow.
(You know that some really smart people answer questions on math.stackexchange, right?)
As was noted in Robert's answer, one is investigating the sequence $x\_{n+1} = | \log(x\_n)|$, whic... | 10 | https://mathoverflow.net/users/nan | 80226 | 48,201 |
https://mathoverflow.net/questions/80186 | 8 | It is a fact that the symmetric groups have as many 2-Sylow subgroups as possible. More precisely, for all $n \geq 1$, the number of 2-Sylow subgroups in $S\_n$ is exactly $n!/2^{\nu\_2(n!)}$, which is the index of a 2-Sylow subgroup of $S\_n$. This follows from (or, depending on which direction you're coming from, pro... | https://mathoverflow.net/users/19012 | Many p,q-Sylow subgroups | The answer is **no**: see Corollary 1.3 in
>
> Robert M. Guralnick; Gunter Malle; Gabriel Navarro, *[Self-normalizing Sylow subgroups](http://www.ams.org/journals/proc/2004-132-04/S0002-9939-03-07161-2/home.html)*, Proc. Amer. Math. Soc. **132** (2004), 973-979.
>
>
>
| 11 | https://mathoverflow.net/users/430 | 80235 | 48,206 |
https://mathoverflow.net/questions/80227 | 4 |
>
> I'm looking for a specific construction, taking an abelian group (with designated element) $(G,+,1)$ to a commutative ring $(R,+,\cdot,1)$, where $G\subset R$ as a pointed abelian group, and which is universal in the following sense: for any commutative ring $(S,+,\cdot,1)$ and any map $f:G\rightarrow S$ preservi... | https://mathoverflow.net/users/15735 | Constructing a ring from an abelian group in a minimal way | Given an abelian group $A$ with a fixed element $e\in A$, you can construct the universal map $f$ from $A$ to a (commutative or noncommutative, as you prefer) ring $R=R(A,e)$ such $f(e)$ is the unit element in $R$. Just take the symmetric algebra $S(A)$ (if you want a commutative ring) or the tensor algebra $T(A)$ (if ... | 8 | https://mathoverflow.net/users/2106 | 80240 | 48,209 |
https://mathoverflow.net/questions/80198 | 1 | Let $M$ and $N$ be $R$-modules for some ring $R$. There is a standard result involving the computation of $\text{Ext}^n(M,N)$, using projective resolutions, which says that you can always choose a projective resolution such that the maps you get from the projective resolution to the extension are always surjective (thi... | https://mathoverflow.net/users/19048 | Can injective resolutions be 'enlarged' (or shrunk) to admit only injective maps from extensions? | I'm afraid the answer is NO! At least in the full generality in which the question is stated, there is pretty easy to disprove it.
Let consider the standard injective resolution of the abelian group $\mathbf Z$:
$$0\to{\mathbf Z}\to{\mathbf Q}\to\mathbf{Q}/{\mathbf Z}\to 0$$. Now if
$$0\to{\mathbf Z}\to E^0\to E^1\... | 2 | https://mathoverflow.net/users/15541 | 80244 | 48,210 |
https://mathoverflow.net/questions/80228 | 15 | What is known about compressing graphs? Here, with "compressing", I mean something like "putting a graph into a zip program"; or with a more technical expression, what is know about the Kolmogorov complexity of a graph? Does it make sense to define something in this line at all? I guess one needs a binary string, first... | https://mathoverflow.net/users/nan | Compressing Graphs (Kolmogorov complexity of graphs) | Li and Vitányi in their standard textbook on Kolmogorov complexity (3rd edition, p.456) observe
>
> Almost all strings have high complexity. Therefore, almost all tournaments and almost all undirected graphs have high complexity.
>
>
>
This is made more precise in Section 6.4. In particular, they show that the... | 12 | https://mathoverflow.net/users/7252 | 80252 | 48,216 |
https://mathoverflow.net/questions/80243 | 7 | I'm a physical chemist and I am involved in “colloidal dice”. These are small, cube-like particles with a really nice, regular shape. These particles are not really cubic, but more rounded, much like a dice. I've got a neat way to quantify their size and "roundness" and I'm interested in their volume and surface area. ... | https://mathoverflow.net/users/19063 | Surface area of superellipsoid (dice) | You can't expect a closed formula for this surface area. The perimeter of an ellipse, much less the perimeter of a superellipse or the surface area of an ellipsoid or a superellipsoid, is already an integral that doesn't have a formula in the usual sense of an elementary formula. Instead, people did what they always do... | 10 | https://mathoverflow.net/users/1450 | 80256 | 48,218 |
https://mathoverflow.net/questions/79841 | 8 | Let G be an undirected graph with the node set $V$ and the Laplacian matrix $L$. Let $N(v)$ denote the neighbors of a node $v$ and $|N(v)|$ its degree. Then a partition $\pi=(V\_1, V\_2, \ldots, V\_k)$ is almost equitable if it holds that $\forall i \ne j\in\{1,\ldots,k\}$ $\forall v, u\in V\_i$ $|N(v)\cap V\_j|=|N(u)\... | https://mathoverflow.net/users/18968 | Eigenvectors and partitions of graphs | **final thoughts** Let us start with a vector (or just specify the distinct entries) and then try to build a graph. With only 2 distinct entries the partition does need to be almost equitable. I think that it is easy to create high irregularity if some entries are 0. Here is an example cobbled up ad hoc (goals three di... | 2 | https://mathoverflow.net/users/8008 | 80266 | 48,224 |
https://mathoverflow.net/questions/80272 | 1 | Here is a question which seems true to me but I can't rigorously show. Suppose $K$ is a compact subset of $\mathbb{R}^n$ such that $\mathbb{R}^n\setminus K$ is connected, does it follow that for any connected open set $U\subset \mathbb{R}^n$ such that $U\supset K$, $U\setminus K$ is also connected?
| https://mathoverflow.net/users/19074 | A question about connectedness in Euclidean space | Yes. Let $C$ be the closed complement of $U$, then by excision of $C$ we have $H\_1(\mathbb{R}, \mathbb{R} - K) = H\_1(U, U - K)$; since $H\_1(\mathbb{R})=0$, you also have in fact $H\_1(\mathbb{R}, \mathbb{R}-K)= H\_1(U, U-K)= 0$ when $\mathbb{R}-K$ is connected.
So $H\_0(U-K)$ injects into $H\_0(U)$ and $U-K$ must... | 3 | https://mathoverflow.net/users/37021 | 80277 | 48,229 |
https://mathoverflow.net/questions/80280 | 11 | Let $X$ be an abelian variety over an algebraically closed field $k$.
Let $L$ be a line bundle on it equipped with an integrable connection $\nabla: L \rightarrow L \otimes \Omega^1\_{X/k}$.
Does it then automatically folllow that $L$ is a bundle in $Pic^0(X)$?
And how general can one make such a statement?
I mea... | https://mathoverflow.net/users/18183 | Line bundles with integrable connection on abelian varieties | Yes, it is true, though an algebraic proof seems (there may be a simpler proof however) somewhat tricky.
* Such a line bundle lies in $\mathrm{Pic}^\tau(X)$. This is a general fact as a line bundle lies in $\mathrm{Pic}^\tau(X)$ if its rational Chern classes are trivial (this follows from Riemann-Roch) and the Chern ... | 10 | https://mathoverflow.net/users/4008 | 80281 | 48,230 |
https://mathoverflow.net/questions/80220 | 17 | In this topic, I will use the word *uncountable group* referring to groups whose cardinality is $\leq|\mathbb R|$.
**Notation:** $R$ is the hyperfinite $II\_1$-factor, $\omega$ is a free ultrafilter on the natural numbers, $R^\omega$ is the tracial ultrapower, $\tau$ is the unique normalized trace on $R^\omega$, $U(R... | https://mathoverflow.net/users/13809 | Connes' embedding conjecture for uncountable groups | The general situation, where CH fails, may be informed by
the [Keisler-Shelah isomorphism
theorem](https://encyclopediaofmath.org/wiki/Keisler-Shelah_isomorphism_theorem), which
asserts that two first-order structures have isomorphic
ultrapowers if and only if they have the same first-order
theory.
In particular, for... | 11 | https://mathoverflow.net/users/1946 | 80283 | 48,232 |
https://mathoverflow.net/questions/80286 | 14 | Is there something resembling a characterization of which groups can map onto a non-abelian free group? Obviously they cannot have property T, and should have nontrivial abelianization, but are there some positive results?
| https://mathoverflow.net/users/11142 | Groups surjecting onto a free group | Such groups are often called 'very large'. A group with a very large subgroup of finite index is called 'large'. Here are some miscellaneous facts:
* Baumslag and Pride showed that every group of deficiency two (ie with a presentation with two more generators than relators) is large.
* One can deduce from Wise's resi... | 13 | https://mathoverflow.net/users/1463 | 80287 | 48,233 |
https://mathoverflow.net/questions/80289 | 0 | Let $X$ and $Y$ be topological spaces. Assume $Y$ is [contractible](http://en.wikipedia.org/wiki/Contractible_space) (hence, path- connected).
Let $f,g: X \to Y$ be continuous maps. At any fixed $x\in X$, there is a path $P\_x: [0,1]\to Y$ from $f(x)$ to $g(x)\in Y$ such that $P\_x(0)=f(x)$ and $P\_x(1)=g(x)$
We d... | https://mathoverflow.net/users/6770 | Continuity of a homotopy-like function | This proof cannot work since then every two maps $X \to Y$, where $Y$ is path-connected, are homotopic - which is false. On the other hand, if $Y$ is contractible, then every map $X \to Y$ is homotopic to a constant map (since this true for the identity $X \to X$), thus every two maps $X \to Y$ are homotopic.
Even if... | 3 | https://mathoverflow.net/users/2841 | 80290 | 48,234 |
https://mathoverflow.net/questions/80288 | 16 | I have a surjective morphism $\pi: Y \to X$ between smooth projective varieties of the same dimension over some algebraically closed field $k$. Debarre claims in his book "Higher-Dimensional Algebraic Geometry" (1.41) that there is an effective divisor $R$ such that
$$ K\_Y \equiv \pi^\* K\_X + R$$
if $K(X) \subset K(X... | https://mathoverflow.net/users/14385 | Pullback of the canonical divisor between smooth varieties | **EDIT** Originally I claimed a more general statement and along the incremental generalizations I reached a statement that was not true. Thanks to Carlos for pointing out this error! So, I thought it would be fair to point out where the error lied.
The main issue is that the original proof works for a finite morphism,... | 16 | https://mathoverflow.net/users/10076 | 80312 | 48,244 |
https://mathoverflow.net/questions/29090 | 18 |
>
> **Question.** Is there a direct construction of the integers which does not involve taking any quotients?
>
>
>
I am of course aware of the [usual construction](http://en.wikipedia.org/wiki/Integer#Construction). I am also aware of the nice [axiomatic characterization](https://mathoverflow.net/questions/231... | https://mathoverflow.net/users/3993 | Direct construction of the integers | Informally speaking, taking the limit of [two's complement](http://en.wikipedia.org/wiki/Two%27s_complement) as the
number of bits goes to $\infty$,
the integers are just the eventually constant binary sequences (which
are naturally represented by finite binary sequences).
For this to work, said sequences must start w... | 16 | https://mathoverflow.net/users/12106 | 80315 | 48,245 |
https://mathoverflow.net/questions/80321 | 6 | More precisely, if $\mathcal F\_i$ is a system of sheaves, is it the case that
$$
(\lim \mathcal F\_i)\_p = \lim ((\mathcal F\_i)\_p)
$$
and similarly for colimits? I can see how to get a map
$$
(\lim \mathcal F\_i)\_p \rightarrow \lim ((\mathcal F\_i)\_p)
$$
by taking the stalks in the diagram for $\lim \mathcal F\... | https://mathoverflow.net/users/19088 | Is the stalk of the (co)limit of sheaves equal to the (co)limit of the stalks? | Let $F$ be a sheaf on $X$ and $p \in X$. Then $F\_p$ is just the pullback $i^{-1} F$, where $i : \{p\} \to X$ is the inclusion of a point. Now $i^{-1}$ is left adjoint to $i\_\*$, thus cocontinuous, i.e. preserves all colimits. This shows that the canonical morphism $\mathrm{colim}\_i(F\_p) \to \mathrm{colim}\_i(F)\_p$... | 9 | https://mathoverflow.net/users/2841 | 80322 | 48,248 |
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