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/69147 | 5 | Assume $G$ is a profinite group such that the Jordan-Hölder factors appearing in the finite quotients vary in a finite number of isomorphism classes of simple groups. Assume also $G$ to have a finite number of subgroups whose corresponding quotient is simple. Does this imply that $G$ is (topologically) finitely generat... | https://mathoverflow.net/users/3680 | Is a profinite group with a finite number of simple quotients and Jordan-Hölder factors finitely generated? | I think the answer is no. Fix a nonabelian finite simple group $S$ and a sequence $(m\_n)$ of integers at least 2. Define inductively $G\_1=S$ and $G\_{n}=S^{m\_n}\wr G\_{n-1}$. This group admits only $S=G\_1$ as simple quotient and only $S$ as Jordan-Hölder factor. I claim that, provided $(m\_n)$ grows fast enough, th... | 5 | https://mathoverflow.net/users/14094 | 84921 | 50,608 |
https://mathoverflow.net/questions/84692 | 3 | Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that
$|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle.
Also, we know some other graphical representations of the value of $c(G,u)$.
1) Do we have any graphical representation for $|c(G,2)|$?... | https://mathoverflow.net/users/19885 | Graphical representation of chromatic polynomial | The answer to your second question appears to be "no".
As the multiplicity of 0 is the number of connected components of a graph, and for a connected graph the multiplicity of 1 is the number of blocks, then we might hope that for a 2-connected graph, the multiplicity of "2" would be related to the number of 3-connec... | 3 | https://mathoverflow.net/users/1492 | 84931 | 50,615 |
https://mathoverflow.net/questions/84930 | 1 | Hi,
Is it true that covariant derivative on any vector bundle over a manifold X comes from some connection on the bundle of linear frames of that vector bundle? This statement is true for the case of tangent bundle, but I am not sure if it is true in general or not. If it is true then can someone please suggest some re... | https://mathoverflow.net/users/20369 | Covariant derivative | If $E \to X$ is a (finite-dimensional) vector bundle and $P$ is the principal $GL(n)$ bundle of frames, then there is a one-to-one correspondence between covariant derivatives on $E$ and principal connections on $P$. A good reference for details is proposition 4.4 of Lawson and Michelson's ``Spin Geometry" (unfortunate... | 2 | https://mathoverflow.net/users/4622 | 84933 | 50,617 |
https://mathoverflow.net/questions/83749 | 3 | I'm using notation close to Street-Walters *"Yoneda structures"*.
For any locally small category $\textbf{A}$
there are, of course, $\hat{\textbf{A}}:=\textbf{set}^{\textbf{A}^{op}}$ and
$\check{\textbf{A}}:=(\textbf{set}^{\textbf{A}})^{op}$
as well as the corresponding Yoneda embeddings
$Y(\textbf{A}):\textbf... | https://mathoverflow.net/users/20027 | The contravariant side of the Yoneda stucture of Cat | I realize that the question was not precise, but I now hope to understand.
For any locally small functor $F:\textbf{A}\rightarrow\textbf{B}$, evaluation of $F$ on arrows can be encoded via $\chi^F:Y(\textbf{A})\Rightarrow\textbf{B}(F,1)F$ or $\psi^F:\textbf{B}\langle 1,F \rangle F \Rightarrow Z(\textbf{A})$.
SW ... | 0 | https://mathoverflow.net/users/20027 | 84940 | 50,621 |
https://mathoverflow.net/questions/84887 | 0 | In order to be able to use a basic possibility function as a Body of Evidence in the [Dempster-Shafer Theory](http://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory) of Evidence, it is needed to transform the function to its Möbius representation.
There is a transformation for discrete possibility functions whic... | https://mathoverflow.net/users/20360 | Möbius Transform of a Continuous Possibility Function | (edit) Okay, so as far as i can see you want to find a replacement for the mobius transform, but for a $\sigma$-algebra. In fact I'm going to guess that your $\sigma$-algebra is the measurable sets in the unit interval, based on what you've said.
The most general setting I know of in which you can define a Möbius fun... | 1 | https://mathoverflow.net/users/20281 | 84941 | 50,622 |
https://mathoverflow.net/questions/84521 | 21 | I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is [*From calculus to cohomology*](https://books.google.com/books?id=CwQ-L9MOGUwC&lpg=PP1&pg=PP9#v=onepage&q&f=false) by Madsen & Tornehave. I know the statement of the theorem is as follows... | https://mathoverflow.net/users/7780 | On the generalized Gauss-Bonnet theorem | When $E\to M$ is an oriented vector bundle of rank $2n$ over a
compact manifold $M$, it has a well-defined *de Rham Euler class* $e(E)$
in $H^{2n}\_{dR}(M)$, and a representative $2n$-form for $e(E)$
can be computed as follows:
Fix a positive definite inner product $\langle,\rangle$ on $E$. (Since any two such in... | 32 | https://mathoverflow.net/users/13972 | 84960 | 50,629 |
https://mathoverflow.net/questions/84948 | 1 | Consider $F$ a non archimedean field and let $o$ be its ring of integer
Let $B$ be the Iwahori subgroup of $GL\_n(F)$ (resp. $GL\_n(o)$) and let $N$ be the normalizer of the diagonal matrices (respective the diagonal matrices).
$B$ and $N$ give a $BN$ pair for $GL\_n(F)$. Is there an explicit algorithm on the group... | https://mathoverflow.net/users/10400 | Algorithm for the cell multiplication rule for GL(n,F) | The asserted cell multiplication isn't quite right as it stands. First, GL(n) does not have "strict" BN-pair structure, but SL(n) does. An obvious extra element needs to be added for GL(n).
Second, for the strict BN-pair situation of SL(n,F) and SL(n,o), the cell multiplication rules are all generated by two cases o... | 4 | https://mathoverflow.net/users/15629 | 84961 | 50,630 |
https://mathoverflow.net/questions/84936 | 2 | Can I have some examples of **finite non-commutative connected** group schemes over a field $k$?
I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks.
| https://mathoverflow.net/users/11964 | finite non-commutative local group schemes | If $\mathrm{char}(k)=p>0$ and $G$ is a $k$-group scheme of finite type, the kernel of the relative frobenius $F\_{G/k}:G\to G^{(p)}$ is a finite connected $k$-group scheme. It has the same Lie algebra as $G$, and in particular it is noncommutative if the Lie algebra is nonabelian, e.g. for $G=GL\_{n,k}$, $n\geq2$.
I... | 6 | https://mathoverflow.net/users/7666 | 84965 | 50,632 |
https://mathoverflow.net/questions/84865 | 8 | I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A\_{n\times n}\sim\mathbb{W}(\Sigma\_{n\times n},m)$ where $\mathbb{W}(\Sigma\_{n\times n},m)$ denotes the [Wishart distribution](http://en.wikipedia.org/wiki/Wishart_distribution) with covariance $\Sigma\_{n\times n}$ and de... | https://mathoverflow.net/users/nan | Eigenvalue distributions of finite dimensional Wishart matrices | Hi, I think you should have a look at this:
Zanella, A., M. Chiani and M.Z. Win,
"On the marginal distribution of the eigenvalues of wishart matrices"
IEEE Transactions on Communications 57 (2009):1050-1060
Cheers, FP
| 0 | https://mathoverflow.net/users/20380 | 84968 | 50,634 |
https://mathoverflow.net/questions/84963 | 1 | I am trying to look at a representation (so a homomorphism) of a group G, and see what the restriction of the representation to a subgroup of G will be. Is there an easy way (or any way!) to do this in MAGMA?
| https://mathoverflow.net/users/19783 | Homomorphisms and their restrictions in MAGMA | If your representation R is of type Map (which it will be if you defined it as Representation(M) for a G-module M), then to restrict R to subgroup H
RH := map< H->Codomain(R) | x :-> R(x) >;
should work.
If you have defined R as a group homomorphism G -> GL(n,K) for some field K, then you could instead use
RH :... | 3 | https://mathoverflow.net/users/35840 | 84972 | 50,637 |
https://mathoverflow.net/questions/84973 | 1 | Is there any example (or more ambitiously, classification) of $X$ with following properties?
* $X$ is a variety over $\mathbb{C}$;
* $X$ is projective and normal;
* $\rho(X) = 1$;
* $X$ is birational to $\mathbb{P}^n$.
Also, I want to hear a result after adding a singularity condition:
How about when $X$ is $\mat... | https://mathoverflow.net/users/4643 | Example of rational projective variety of Picard number 1 | Hyperquadrics of dimension at least three.
| 4 | https://mathoverflow.net/users/605 | 84976 | 50,639 |
https://mathoverflow.net/questions/84975 | 4 | Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the resulting 3-manifold have a hyperbolic structure with finite volume? I am tempted to say 'no' because it would have an end wh... | https://mathoverflow.net/users/20382 | Mapping torus relative to an infinite orbit can be hyperbolic with finite volume? | An end of an orientable finite volume hyperbolic $3$--manifold always has a neighborhood homeomorphic to $S^1 \times S^1 \times \mathbb{R}$, so no. Introductory texts on hyperbolic manifolds will contain this result.
In your case, you could simply check that any neighborhood of the end has infinite volume.
| 6 | https://mathoverflow.net/users/1335 | 84990 | 50,645 |
https://mathoverflow.net/questions/84971 | 3 | Hi,
given a triple of spaces $(X,A,U)$, that is excisive with respect to some homology theory $H$, is the triple $(SX,SA,SU)$ again excisive?
Here SY means unreduced suspension of Y, and there's an obvious identfication in making $(SX,SA,SU)$ a triple.
By being excisive I mean that the inclusion gives an isomorph... | https://mathoverflow.net/users/17462 | Suspension of an excisive pair | No. Suppose that $X$ is an $n$-sphere, $A$ is a closed hemisphere, and $U$ is a point in the interior of $A$. Then $SX$ is an $(n+1)$-sphere, $SA$ is a closed hemisphere, and $SU$ is a closed arc in $SA$ with endpoints in the boundary. $SX-SU$ is contractible and $SA-SU$ is homotopy-equivalent to an $(n-1)$-sphere, so ... | 4 | https://mathoverflow.net/users/6666 | 84991 | 50,646 |
https://mathoverflow.net/questions/84950 | 4 | Suppose that $X=\bigcup\_{n=1}^\infty K\_n$ is a topological space, $K\_n$ is a metrizable subspace in $X$ for every $n \in \omega$, then $X$ is a metrizable space?
In metrizable spaces, compactness is equivalent to $\sigma$-compactness?
**One more:** Is pseudocompactness hereditary with respect to $\sigma$-compact... | https://mathoverflow.net/users/18465 | Are countable unions of metrizable spaces metrizable too? | No to all your questions. There are lots of countable non-metrizable spaces: an easy one is $\mathbb{N}$ in the cofinite topology, which can be written as a countable (disjoint) union of singletons (which are metrizable, and closed and compact). Using ultrafilter spaces (given an ultrafilter $\mathcal{F}$ on $\mathbb{N... | 9 | https://mathoverflow.net/users/2060 | 84995 | 50,649 |
https://mathoverflow.net/questions/85000 | 0 | Hello,
I have come across the function
$f(t) = \sum\_{j=1}^n c\_j e^{2 \pi i a\_j t}$
with $c\_j \in \mathbb{C}$, $c\_j\neq 0$ and $a\_j\in\mathbb{R}$, $a\_j \neq 0$ for $j=1,...,n$, and the $a\_j$ distinct. I want to show that $f(t)$ is periodic with least period equal to $1/\gcd a\_j$ if the $a\_j$ have a commo... | https://mathoverflow.net/users/20381 | Least common period of a finite sum of exponentials | For functions of this form, define an inner product by $\langle f,g\rangle=\lim\_{T\to\infty}\frac1T\int\_0^T f(t)\bar g(t)\,dt$. With this inner product, the set of functions $e^{2\pi i at}$ form an uncountable orthogonal set.
If $f(t)$ is periodic with period $s$, then $f(t)=f(t+s)=\sum\_{j=1}^n (e^{2\pi i a\_js}c\... | 2 | https://mathoverflow.net/users/11054 | 85004 | 50,652 |
https://mathoverflow.net/questions/85006 | 3 | Consider the set $\mathcal{P}(\mathbb{R})$ of all subsets of $\mathbb{R}$, the set of real numbers. It has a natural partial order: $A \leq B$ iff $A \subseteq B$.
Can one extend this order to a total order?
(I was discussing this with a friend and we didn't know if this is possible. If we replace $\mathbb{R}$ by a... | https://mathoverflow.net/users/7313 | Extensions of the partial order of the power set | Michael Greinecker's answer (in a comment) is correct in the context of the usual axioms of set theory, including the axiom of choice. If one works in set theory without the axiom of choice, then one cannot prove that $\mathcal P(\mathbb R)$ admits any total order at all (even without the requirement that it extend $\s... | 10 | https://mathoverflow.net/users/6794 | 85009 | 50,653 |
https://mathoverflow.net/questions/85008 | 3 | Consider a sequence of complex valued measures \mu\_{n} in the euclidean space \R^d which converges weakly to some compactly supported measure \mu. The weak convergence is in the sens that \int\_{\R^d} \psi d\mu\_n converges to \int\_{\R^d} \psi d\mu for each smooth function with compact support $\psi$.
My problem is... | https://mathoverflow.net/users/14436 | extension of the convergence of a sequence of measures | No; here's an easy counterexample. Let $\mu\_n$ be the uniform measure on the interval $[n,n+1]$. This sequence of compactly supported measures converges weakly to the zero measure, in the sense you described, because the supports of the $\mu\_n$'s eventually move away from the compact support of your $\psi$. Furthermo... | 5 | https://mathoverflow.net/users/6794 | 85011 | 50,655 |
https://mathoverflow.net/questions/84977 | 12 | This question has been "manually migrated" to TeX-SX: <https://tex.stackexchange.com/q/40200/86>
---
Apologies if the question is not very appropiate for Mathoverflow. It seems to me more appropiate here than in the other 'exchange' sites.
**Is there an IT tool to create a graph of dependencies from a Latex fil... | https://mathoverflow.net/users/1887 | Graph of dependencies from a Latex file | As a rule, you cannot depend upon math papers making every dependency explicit, meaning you cannot extract nearly so much information from this directed graph as you imagine. In addition, there isn't any reason this graph should be acyclic since forward references frequently get used in outlines and motivational text.
... | 4 | https://mathoverflow.net/users/14163 | 85014 | 50,656 |
https://mathoverflow.net/questions/85012 | 0 | Given a line function $y = ax + b$, it is easy to calculate the sum-of-squares distance between the line and a window of samples $(1, y\_1), (2, y\_2), ..., (n, y\_n)$ (where $y\_1$ is the oldest sample and $y\_n$ is the newest):
$\sum\_{x=1}^{n}(y\_x - (ax + b))^2 $
I need a fast algorithm for calculating this val... | https://mathoverflow.net/users/11998 | Algorithm for calculating the sum-of-squares distance of a rolling window from a given line function | We have $\sum\_x (y\_x - ax-b)^2 = \sum\_x y\_x^2 - 2a \sum\_x x y\_x - 2b \sum\_x y\_x + \sum\_x (ax+b)^2$ so the only term requiring $O(n)$ time per shift is $\sum\_x x y\_x$ because an easy $O(1)$ time trick handles the other terms involving $y\_x$.
In this term, you can decrement $x$ in $O(1)$ time too because $\... | 1 | https://mathoverflow.net/users/14163 | 85018 | 50,659 |
https://mathoverflow.net/questions/85021 | 1 | I have a question regarding enumeration of perfect lattices/quadratic forms. In the thesis of Achill Schürmann <http://fma2.math.uni-magdeburg.de/~achill/public/habil.pdf> there is an algorithm called "Voronoi's algorithm" in chapter 3, used to enumerate arithmetically inequivalent perfect quadratic forms (meaning the ... | https://mathoverflow.net/users/18693 | Enumerating Perfect Lattices | The algorithm stops when you don't get any more perfect forms. Specifically, at each step you determine all the contiguous forms $Q\_i$ and test whether they are equivalent to forms you already knew. For all the ones that aren't, you add them to the list and iterate to determine all the forms contiguous to them, etc., ... | 2 | https://mathoverflow.net/users/4720 | 85025 | 50,663 |
https://mathoverflow.net/questions/84989 | 8 | There are many results on the spacing of the gaps between nontrivial zeros of the $\zeta$ function, from trivial (average value is $\frac{2\pi}{\log\gamma\_n}$) to difficult (bounds on max and min values of the normalized gap). Are any reasonable upper bounds known? I'd like to have something that says, given any $\var... | https://mathoverflow.net/users/6043 | Upper bounds on the difference of consecutive zeta zeros | Littlewood was the first to prove that the gaps between the ordinates of successive zeros of $\zeta(s)$ tend to zero. This is proved, for instance, in Titchmarsh's book on the zeta-function (see Theorem 9.11).
I believe the best known unconditional result states that
$$ \gamma\_{n+1}-\gamma\_n = O( 1/\log\log\log \ga... | 14 | https://mathoverflow.net/users/3659 | 85029 | 50,667 |
https://mathoverflow.net/questions/85013 | 43 | Let
$$ c\_n = \sum\_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$
It is clear that $c\_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum defining $c\_{2m}$, the sequence $(c\_{2m})$ may be very well behaved.
>
> Is $c\_n > 0$ for all even $n$?
>
>
>
An ... | https://mathoverflow.net/users/7709 | Alternating sum of square roots of binomial coefficients | Here's a proof of the positivity of
$$
c\_n(\alpha) := \sum\_{r=0}^n (-1)^r {n\choose r}^\alpha
$$
for all even $n$ and real $\alpha < 1$. It follows
(via M.Wildon's clever $F(x) F(-x)$ trick at mo.84958) that
$\sum\_{n=0}^\infty \phantom. x^n / n!^{\alpha} > 0$ for all $x \in\bf R$.
[**EDIT** fedja has meanwhile provi... | 46 | https://mathoverflow.net/users/14830 | 85035 | 50,671 |
https://mathoverflow.net/questions/83349 | 1 | I want to show that if $w$ is a $p$-form such that its induced cochain on $p$-chains:
$w(\gamma)= \int\_{\gamma} w \in S$
takes values in a discrete set $S \subset \mathbb{R}$ then $w$ must be zero.
* My idea is to say that we know some chains (i.e. the zero chain) will integrate to zero and that there is a way t... | https://mathoverflow.net/users/19926 | A p-form taking discrete values on p-chains must be 0. | You can determine the value of the $p$-form at a point as the limit if integrals over very small $p$-simplices and rescaling. If the integral takes values in a discrete subgroup of $\mathbb{R}$, then you get zero.
| 3 | https://mathoverflow.net/users/7530 | 85036 | 50,672 |
https://mathoverflow.net/questions/85031 | 1 | Dear all,
I've got a SDP problem as follows:
$\min\_{{\bf H}\succeq0}\quad trace({\bf H}) - {\bf a}^{\top}{\bf H}{\bf b}$,
where ${\bf a}$ and ${\bf b}$ are two constant vectors. May somebody tell me how to solve this SDP problem? Thank you very much in advance.
[Added] Thanks for Suvrit to point out some issue... | https://mathoverflow.net/users/5531 | A positive semidefinite programming problem | Your problem has no solution. Here is why.
Let $H$ be $2 \times 2$. Let $a=(2, 0)$ and $b=(1, 0)$. Then, since $a^THb=\mbox{tr}(Hab^T)$, the objective function of your problem can be rewritten as $\mbox{tr}(H-Hab^T) = \mbox{tr}(HC)$, where
$$C = I-ab^T = \begin{bmatrix} -1 & 0\\\\ 0 & 1\end{bmatrix}.$$
Now you can ... | 2 | https://mathoverflow.net/users/8430 | 85043 | 50,674 |
https://mathoverflow.net/questions/84958 | 100 | Is $$ \sum\_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.) If so, how would one prove this? (To confirm: This is the power
series for $e^x$, except with the denominator replaced by $\sqrt{n!}$.)
| https://mathoverflow.net/users/20242 | Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive? | Looks like the computers really spoiled us :)
GH gave a perfectly valid answer already but the cheapest way to prove positivity is to write $\int\_0^1(1-t^n)\log(\frac 1t)^{-3/2}\,\frac{dt}t=c\sqrt n$ with some positive $c$ (just note that the integral converges and the integrand is positive, and make the change of ... | 121 | https://mathoverflow.net/users/1131 | 85048 | 50,677 |
https://mathoverflow.net/questions/84944 | 1 | If $G$ is a group scheme of finite type over a field $k$, then one can study it's Hopf Algebra if it is affine. This is clear, but now if $G$ is not affine, one seems to do the following: complete the local ring $\mathcal O\_{G,e}$ of the zero point with respect to it's maximal ideal and then one gets a comultiplicatio... | https://mathoverflow.net/users/18183 | Question about formal group schemes | Show your product on $G$ restricts to a product on formal neighborhoods of the identity (via the Hopf algebra correspondence you mentioned these are coalgebra structures on the quotients of the powers of the maximal ideal of $\mathcal O\_{G,e}$), these small group schemes form a directed system, then take the associate... | 1 | https://mathoverflow.net/users/8818 | 85050 | 50,679 |
https://mathoverflow.net/questions/85052 | 1 | Let $(M,g)$ be a closed riemannian surface . let $\alpha$ be a simple closed geodesique . does there is exist a simple closed geodesic $\beta$ that intersect alpha at only 1 point p such that $[\alpha]$ and $[\beta]$ does not commute in $\pi\_1(M,p)$
| https://mathoverflow.net/users/nan | intersection of geodesiques | Yes. Consider the punctured torus, then the $(1, 0)$ and $(0, 1)$ curves together generate the fundamental group (which is the free group on two generators), and so don't commute. Now, if you have a *closed* riemann surface, one of its handles is a punctured torus, so the above construction goes through without change.... | 1 | https://mathoverflow.net/users/11142 | 85053 | 50,680 |
https://mathoverflow.net/questions/85056 | 1 | Perhaps this is not the right place to ask the following question but I did not find any suitable on the web. So I would be very grateful for sharing your experience.
What is a good program to draw cobordisms or surfaces (oriented) in order to integrate them in a Latex file?
I have the same question for graphs and ... | https://mathoverflow.net/users/20405 | Program for drawing cobordisms | Planar stuff is not too hard. I generally use the LaTeX package tikz, because that way everything is in the LaTeX file.
Three-dimensional pictures are harder. If you know equations for your surfaces then you can plot them using Maple or Mathematica and then export as jpeg say, and then include in your LaTeX file usin... | 4 | https://mathoverflow.net/users/10366 | 85057 | 50,681 |
https://mathoverflow.net/questions/85055 | 7 | Given a matrix $A\in \mathbb{R}^{n\times n}$, I am looking for
a symmetric matrix $S\in\mathbb{R}^{n\times n}$ such that
$$
S A + A^T S = I
$$
$A$ can be assumed to be regular (with positive determinant, if this is of any help).
The difficulty is of course that $S$ must be symmetric, otherwise one could simply t... | https://mathoverflow.net/users/6035 | A Linear Algebra Problem | These matrix equations are called *Lyapunov equations* and are extensively studied in control theory.
For instance, if $A$ is Hurwitz (all eigenvalues in the left half-plane), then the unique symmetric solution of $A^TX+XA+Q$ is
$$
X=\int\_0^\infty e^{A^T t } Q e^{At} dt.
$$
| 11 | https://mathoverflow.net/users/1898 | 85058 | 50,682 |
https://mathoverflow.net/questions/85051 | 4 | I was considering the following game on an undirected unweighted graph $G=(V,E)$ (not necessarily simple). Two players, Police and Runaway, take moves in turn. Police can cut an arbitrary subset of edges in a single move and Runaway can move from a current vertex to an adjacent vertex (cutting an edge means that corres... | https://mathoverflow.net/users/20404 | Graph connectivity related game | This looks like problem J (titled "Tunnels") from the 2007 edition of a computing olympiad called the [ACM ICPC](http://cm.baylor.edu/welcome.icpc); the problem statement is [here](http://cm.baylor.edu/ICPCWiki/attach/Problem%20Resources/2007WorldFinalProblemSet.pdf) and the problemsetter's solution is mirrored in [thi... | 2 | https://mathoverflow.net/users/16139 | 85070 | 50,689 |
https://mathoverflow.net/questions/85065 | 16 | I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof strategy works out, because I think such a collection of examples would be nice for impressing students of high school alge... | https://mathoverflow.net/users/1106 | Unexpected applications of the fact that nth degree polynomials are determined by n+1 points | If you want research level mathematics, the joint theorem is an excellent example of the polynomial technique that can be presented to high-school students. The statement is
$n$ lines in the space can form at most $Cn^{3/2}$ joints (the points where at least three non-coplanar lines intersect).
The proof (for an ex... | 11 | https://mathoverflow.net/users/1131 | 85071 | 50,690 |
https://mathoverflow.net/questions/85068 | 10 | If $\mathbf{C}$ is a category, then the *Yoneda functor* which sends $a$ to $Hom\_\mathbf{C}(-,a)$ is a fully faithful embedding of categories
$$ \mathbf{C}\rightarrow \mathbf{Func}(\mathbf{C}^{op},\mathbf{Set})$$
Given any subcategory $\mathbf{B}\subseteq \mathbf{C}$, there is a similar functor
$$ \mathbf{C}\righta... | https://mathoverflow.net/users/750 | Subcategories which still give a Yoneda embedding | Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of *all* morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of t... | 14 | https://mathoverflow.net/users/2841 | 85081 | 50,697 |
https://mathoverflow.net/questions/85087 | 1 | A proper morphism is defined as separated, of finite type and universally closed. I wonder if the requirement of being of finite type is superfluous, i.e. if being not of finite type implies not universally closed. Recall that a morphism is of finite type if and only if it is locally of finite type and quasi-compact.
... | https://mathoverflow.net/users/2234 | not locally of finite type implies not universally closed? | Let $k$ be a field, $A=k[X\_1,X\_2,\dots]$ and $I=(X\_1,X\_2,\dots)$. Then $\mathrm{Spec}(A/I^2)\to\mathrm{Spec}(k)$ is a universal homeomorphism, but not locally of finite type.
added in edit: In particular, there is no purely topological condition which implies locally finite type.
| 4 | https://mathoverflow.net/users/2035 | 85091 | 50,702 |
https://mathoverflow.net/questions/85067 | 8 | I am looking for information about the symplectic groups $Sp\_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \qquad
f=\begin{pmatrix}0&1\\-1&0\end{pmatrix}=e-e^T$$
on the vector space $(\mathbb{F}\_2)^{2d}$ equipped with the standard ... | https://mathoverflow.net/users/3680 | Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups | Both of these actions are 2-primitive, so the 1-point stabilizer acts primitively on the remaining points.
The 1-point stabilizers in the two actions are the orthogonal groups ${\rm SO}^{\pm}\_{2m}(q)$,
and the 2-point stabilizers are the maximal parabolic subgroups of these orthogonal groups with structure $2^{2m-2}... | 14 | https://mathoverflow.net/users/35840 | 85094 | 50,704 |
https://mathoverflow.net/questions/85089 | 6 | The exciting question on [alternating sums of binomial coefficients](https://mathoverflow.net/questions/85013/alternating-sum-of-square-roots-of-binomial-coefficients) triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be overlooking something ob... | https://mathoverflow.net/users/8430 | Alternating sums of GCDs | One can compute this sum explicitly. Let $n+1=2^a \prod\_i p\_i^{\alpha\_i}$ with $a\geq 1$, then we have:
$$\sum\_{i=1}^n (-1)^{i-1}\mbox{gcd}(n+1,i)=(-1)^{n-1}(n+1)-a2^{a-1}\prod\_i\left((\alpha\_i+1)p\_i^{\alpha\_i}-\alpha\_ip\_i^{\alpha\_i-1}\right)$$
as was proved in
>
> Laszlo Toth, ["Weighted Gcd-Sum Functi... | 9 | https://mathoverflow.net/users/2384 | 85099 | 50,707 |
https://mathoverflow.net/questions/85100 | 0 | Dear Mathoverflow'ers,
I am interested in the following equation:
$-\Delta u = u^{p-1} + \lambda u$ in $ \Omega$ with $ u=0 $ on $ \partial \Omega$.
1) My question is related to the Brezis-Nirenberg result from 1983 which states (and I am probably slightly off here) that when $ p=2^\*$ (the critical Sobolev ex... | https://mathoverflow.net/users/19597 | Brezis-Nirenberg result compared to abstract bifurcation theory | There are several reasons why the work of Brezis-Nirenberg was surprising.
1. First, it goes beyond a small range of $\lambda$'s that one would obtain from bifurcation theory.
2. The existence of positive solutions is highly dependent on the geometry and topology of $\Omega$.
| 2 | https://mathoverflow.net/users/20302 | 85101 | 50,708 |
https://mathoverflow.net/questions/85095 | 3 | Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not *hyperstonean*). Everyone points out the following reference:
J. Dixmier, Sur certains espaces consideres par M. H. Stone, *Summa Bras. ... | https://mathoverflow.net/users/20412 | ED compact $K$ such that $C(K)$ is not a dual Banach space | Your desired space is discussed in the book "Topics in Banach Space Theory", by Albiac and Kalton. Springer 2006. See Remark 4.3.9, p. 85 and Problems 4.8 and 4.9, p. 99.
| 3 | https://mathoverflow.net/users/20300 | 85106 | 50,711 |
https://mathoverflow.net/questions/85104 | 18 | I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant:
1) Is eta a topological invariant (or geometric invariant)?
2) Which is its relation with the three dimensional Chern-Simons form?
3) In how many no... | https://mathoverflow.net/users/19938 | Atiyah-Patodi-Singer Eta invariant and Chern-Simons form | 1) The eta invariant itself depends on the metric, but the *relative* eta invariant is in many cases (see comments) a homotopy invariant. The relative eta invariant is defined to be the difference of the eta invariants associated to the Dirac operator twisted by two different flat Hermitian bundles (i.e. unitary repres... | 20 | https://mathoverflow.net/users/4362 | 85113 | 50,715 |
https://mathoverflow.net/questions/85115 | 0 | I have a strong feeling that, for a compact connected Riemann surface $X$ of genus $g>0$, the Euler characteristic of the Weierstrass divisor $W$ equals $$\chi(X,\mathcal{O}\_X(W)) = (g-1)^2.$$ Is this true?
Answer:
By Riemann-Roch, the Euler characteristic is given by $$ \chi(X,W) = g^3 -g + 1- g = g^3-2g+1.$$
T... | https://mathoverflow.net/users/20417 | Euler characteristic of Weierstrass divisor | I'm not sure I've heard the term Weierstrass divisor before, but I take it you mean the sum of the Weierstrass points, with multiplicities given by the weights. In this case, the sum of the weights, and hence the degree of the divisor, is given by
$$\sum\_{p\in X} w(p) = (g-1)g(g+1).$$
By Riemann-Roch, the Euler ch... | 0 | https://mathoverflow.net/users/7399 | 85116 | 50,716 |
https://mathoverflow.net/questions/85121 | 3 | If $X$ is a smooth projective curve over a field $k$, $V$ is a subbundle of some finite direct sum of $O\_X$, then is it true that $V$ is of degree 0?
| https://mathoverflow.net/users/18380 | degree 0 vector bundles | No. Counterexample: On $\mathbf P^1$, take a (two-element) basis of $\Gamma(\mathbf P^1, \mathcal O(1))$, so that the corresponding homomorphism $\mathcal O^2\to\mathcal O(1)$ is surjective. Its kernel is a subbundle of degree $-1$.
| 7 | https://mathoverflow.net/users/2035 | 85122 | 50,718 |
https://mathoverflow.net/questions/85141 | 0 | Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ is the maximum integer among $6, 7, 10, 13, 21, 19, 20, 17$ (which are the sums of each 3 consecutive elements, respecti... | https://mathoverflow.net/users/20425 | Maximum sum of 3 consecutive numbers in a permutation | I am sorry, but this is certainly not a research question, and hence (as far as I understand the purpose of this forum) not a suitable question for mathoverflow. I suppose this thread will be closed within the next few minutes (and rightfully so).
However, since I read the question and started thinking about it, here... | 2 | https://mathoverflow.net/users/8590 | 85145 | 50,728 |
https://mathoverflow.net/questions/74961 | 4 | It is well known that characters of affine Lie algebras have
certain modular properties. For instance, the linear span of all
irreducible characters at a given level must be invariant under a
certain action of $SL(2,\mathbb Z)$. In the case of affine $E\_8$
there is only one irreducible level $1$ representation, the ba... | https://mathoverflow.net/users/13377 | Getting certain modular functions from characters | Since you allow virtual characters you should definitely expect such a thing (due to the general philosophy of writing down Eisenstein series as linear combinations of theta series after Siegel, Weil and others).
Here is an explicit construction. Take the simplest affine Kac-Moody Lie algebra, namely $A\_1^{(1)}$, an... | 5 | https://mathoverflow.net/users/nan | 85165 | 50,739 |
https://mathoverflow.net/questions/84730 | 18 | *Greetings to all* !
Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO.
Let $A\in M\_{2n}(k)$ be an alternate matrix. Say that $A$ is non-singular. It is well-known that there exists an $M\in GL\_{2n}(k)$ such that $A=M^TJM$, where
$$J=\begin{pmatrix} 0\_n & I... | https://mathoverflow.net/users/8799 | Alternate and symmetric matrices | I feel that framing this question in terms of matrices rather than bilinear forms on a vector space obscures what is actually going on and makes it harder to understand what needs to be proved. Here is how I would describe the problem and the partial answer that results from this description:
Let $V$ be a finite dime... | 27 | https://mathoverflow.net/users/13972 | 85166 | 50,740 |
https://mathoverflow.net/questions/85128 | 2 | Suppose $\mathscr{A}$ is a complete Boolean algebra (assume it is c.c.c. if you wish). Is there any, say, *canonical* embedding $\mathscr{A}\subseteq \mathscr{B}$ into a complete Boolean algebra which is weakly countably distributive ($(\omega, \omega)$-distributive)? Note that I do not put any extra assumptions on $\m... | https://mathoverflow.net/users/20412 | Extending BAs to weakly countably distributive algebras. | Let me make a few observations.
First, although you have insisted that the Boolean algebras be complete, there can be no *complete* embedding of $\mathscr{A}$ into a distributive $\mathscr{B}$, if $\mathscr{A}$ did not already exhibit that degree of distributivity, since a complete embedding would preserve any counte... | 3 | https://mathoverflow.net/users/1946 | 85169 | 50,742 |
https://mathoverflow.net/questions/85177 | 4 | Let $\Phi$ be a set of bijections $\phi\_a:X\to Y$. To each pair of bijections $\phi\_a$, $\phi\_b$ one naturally relates a bijection $\psi\_{ab}:=\phi\_a^{-1}\circ\phi\_b: X\to X$. In some cases the set of all such $\psi\_{ab}$ forms a subgroup of $Sym(X)$, the group of all bijections $X\to X$.
Were these kinds of c... | https://mathoverflow.net/users/11100 | a group from a family of bijections X->Y | The name you are looking for is that of a **torsor** or **principal homogeneous space**. For any sets $X$ and $Y$, the set of bijections $X \to Y$ is a torsor for the group $\mathrm{Sym}(X)$ acting on the right as well as the group $\mathrm{Sym}(Y)$ acting on the left. In your case, the set $\Phi$ is a torsor for the a... | 7 | https://mathoverflow.net/users/1310 | 85179 | 50,745 |
https://mathoverflow.net/questions/85171 | 5 | I apologize for the long question. My ultimate aim is to understand the existing terminology and find appropriate terminology for covariant derivatives on vector bundles which generalizes to "covariant" derivatives on fiber bundles (a linear Ehresmann connection is to a [linear] covariant derivative as a nonlinear Ehre... | https://mathoverflow.net/users/19516 | Terminology of "covariant derivative" and various "connections" |
>
> As I understand it, the "covariant"
> part of this comes from the fact that
> the T∗M component changes covariantly
> under coordinate changes and not how
> the E component changes. Is this
> correct?
>
>
>
Yes.
>
> The motivation for the qualifier
> "covariant" seems to ultimately stem
> from coo... | 4 | https://mathoverflow.net/users/20302 | 85182 | 50,746 |
https://mathoverflow.net/questions/85134 | 8 | In Geometric Invariant Theory, by Mumford, Fogarty, and Kirwan, if there is a mention of descent theory, it almost always comes along with a reference to SGA 8, Theorem 5.2 (see the end of the proof of prop 6.9 on page 119, for example). Now I know SGA 8 was never made, but I was wondering:
1. Does anyone have a good... | https://mathoverflow.net/users/18403 | References to SGA 8 and descent theory | For question 1, see the comment above.
Collecting the answers to question 2:
* Grothendieck's original *FGA*, starting with [TDTE I](http://www.numdam.org/numdam-bin/fitem?id=SB_1958-1960__5__299_0)
* Vistoli's chapter in *FGA explained*, for the connection with stacks
* [What is descent theory?](https://mathoverfl... | 5 | https://mathoverflow.net/users/2035 | 85185 | 50,749 |
https://mathoverflow.net/questions/85204 | 10 | This is probably an easy question, but I'm not able to figure it out.
Are the following the same:
1. Field of fractions of the ring of Witt vectors over the algebraic closure of $\mathbb{F}\_p$
2. Algebraic closure of the field of p-adics (which is the field of fractions of the ring of Witt vectors over $\mathbb{F}... | https://mathoverflow.net/users/3040 | Ring of Witt vectors and p-adics | no: the Witt ring of $\bar{F\_p}$ is a complete DVR and so its field of fractions will be a complete local field; but the algebraic closure of $Q\_p$ is not complete.
However, take the maximal unramified extension of $Q\_p$; this is a non-complete field. Its completion $F$ is the fraction field of the Witt ring of $... | 16 | https://mathoverflow.net/users/11786 | 85205 | 50,757 |
https://mathoverflow.net/questions/85212 | 12 | Hey Everyone!
So I am new blood in the topic of Khovanov Homology and related topics. According to my basic reading the idea is to get the Jones polynomial as the Euler Characteristic of a certain homology theory. My question is, why do this? I mean this in the sense that why is it important/interesting to have this "q... | https://mathoverflow.net/users/9187 | Why "Categorify"? Relating to link/knot homologies... | A good reason is that categorified invariants are usually more subtle than the uncategorified ones, and their additional structure gives more information about the knot/link. For example there are rather simple knots with the same Jones polynomial that are distinguished by their Khovanov homology. Furthermore, Kronheim... | 14 | https://mathoverflow.net/users/12952 | 85215 | 50,760 |
https://mathoverflow.net/questions/85199 | 1 | Every proof I've read about this fact considers two cases: $A$ - finite and $A$ - infinite but this is undecidable problem. So, is there constructive proof?
| https://mathoverflow.net/users/19484 | Is there constructive proof of the fact that every recursive set $A \ne \varnothing$ is recursively enumerable in non-decreasing order? | Here is Goldstern's answer, transcribed to constructive mathematics.
In constructive mathematics we do not speak of "recursive" but rather decidable subsets of $\mathbb{N}$. (Recall that a subset $X \subseteq Y$ is decidable if $\forall y \in Y. y \in X \lor y \not\in X$.) Also, your assumption that $A \neq \emptyset... | 11 | https://mathoverflow.net/users/1176 | 85222 | 50,765 |
https://mathoverflow.net/questions/85138 | 6 | We denote by $\frak p\le q$ the abbreviation that there is $f:\frak p\to q$ which is injective, and by $\frak p\le^\ast q$ we abbreviate that there is a surjection from $\frak q$ onto $\frak p$.
If $X$ is a set in a universe of ZF, denote by $H(X)=\min\lbrace\alpha\mid\alpha\nleq X\rbrace$ known as The Hartog number ... | https://mathoverflow.net/users/7206 | Surjective Maps onto $\aleph$-numbers | I originally posted this question in hope that someone else knew of a reference for an answer, however it seemed to me that indeed the best way is to solve this on my own. I tried to imitate Monro's proof, to a certain extent, and I believe that I have succeeded:
$\renewcommand{\Dom}{\operatorname{Dom}}\renewcommand{\H... | 4 | https://mathoverflow.net/users/7206 | 85223 | 50,766 |
https://mathoverflow.net/questions/85230 | 6 | Let $p\_n$ be the n-th prime number and $c\_n$ be the n-th composite number. We have
$$
\lim\_{n \to \infty}\frac{1}{n} \sum\_{r=1}^{n}\frac{p\_n^2}{p\_n^2 + p\_r^2}
= \lim\_{n \to \infty}\frac{1}{n} \sum\_{r=1}^{n}\frac{c\_n^2}{c\_n^2 + c\_r^2} = \frac{\pi}{4}.
$$
The beauty of the above result is that the first ... | https://mathoverflow.net/users/20174 | On prime numbers | The sums above are Riemann sums over different partitions for the function $\arctan(x)$ between $x=0$ and $x=1$, so the limit will equal the integral which is $\frac{\pi}{4}$. Both of these converge to the same value because they are not too weirdly distributed among $[0,1]$.
**Remark:** We need to use the fact that ... | 22 | https://mathoverflow.net/users/12176 | 85231 | 50,770 |
https://mathoverflow.net/questions/85155 | 2 | Hi all.
I have some question on Hecke operators and its relations to the Hecke algebra.
(1)
I want to understand why the "Hecke algebra" is finitely generated in some cases. I found a nice result in W.Stein, Modular Forms, a Computational Approach, Thm 9.23 (see <http://wstein.org/books/modform/modform/newforms.h... | https://mathoverflow.net/users/20431 | "Hecke algebra" finitely generated? | I will address point (1), namely why $\mathbf{T}$ is finitely generated as a $\mathbf{Z}$-module.
We have a pairing $\mathbf{T} \times S\_k(N,\mathbf{Z}) \to \mathbf{Z}$ given by $\langle T,f \rangle = a\_1(Tf)$. It is left-nondegenerate because if for every $f$ we have $\langle T,f \rangle =0$ then we also have $a\_... | 4 | https://mathoverflow.net/users/6506 | 85239 | 50,774 |
https://mathoverflow.net/questions/85242 | 3 | Let $G$ and $H$ be two connected Lie groups. By the Dold-Lashof construction the classifying space $BHom(G,H)$ is well-defined (similar to the Milnor construction).
Is there a relation between $BHom(G,H)$ and the space of pointed maps $Map\_0(BG,BH)$?
More precisely, could there be a homotopy equivalence or highly co... | https://mathoverflow.net/users/20451 | Dold-Lashof construction and classifying space functor | What is $BHom(G;H)$? Typically, $Hom(G;H)$ is not a group unless $H$ is abelian. Maybe you want to talk about the natural map
$$
Hom(G;H) \to Map\_0 (BG;BH)
$$
from the space of homomorphisms to the mapping space. In some cases, this is a homotopy equivalence, for example if $G$ is connected and compact and $H=U(1... | 9 | https://mathoverflow.net/users/9928 | 85245 | 50,776 |
https://mathoverflow.net/questions/85213 | 2 | I am new to this branch of math, so bear with me.
This question started when reading Kevin McCrimmon's "A Taste of Jordan Algebras"
It talks about polarization and gives a general description.
>
> the general process of linearization
> (often called polarization, espe-
> cially in analysis in dealing with
> quadr... | https://mathoverflow.net/users/20445 | polarization/linearization as in jordan forms | Alternatively, you can polarise right away as follows: if $p(x)$ is homogeneous of degree $n$ (here $x$ may be a variable with values in $\mathbb{R}^k$, e.g. $p(x)=\mathop{\mathrm{tr}}(x^4)$, where $x$ is a matrix), then you can look at
$$
p(\lambda\_1x\_1+\lambda\_2x\_2+\cdots+\lambda\_nx\_n),
$$
where $\lambda\_i$... | 2 | https://mathoverflow.net/users/1306 | 85257 | 50,784 |
https://mathoverflow.net/questions/85241 | 4 | What is the asymptotics (in L) for the number of topologically different knots possible using a perfectly flexible, non-selfintersecting rope of length L and radius 1? (With ends glued together after knotting)
| https://mathoverflow.net/users/20435 | Growth of knots possible with rope of length L | I believe this is an open question. There are some possible estimates.
Let $cr(K)$ denote the crossing number of $K$. There are upper and lower bounds on the ropelength in terms of crossing number (see the section "[Dependence of ropelength on other knot invariants](http://en.wikipedia.org/wiki/Ropelength)"). There a... | 6 | https://mathoverflow.net/users/1345 | 85263 | 50,786 |
https://mathoverflow.net/questions/84703 | 4 | Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces?
Note it is well known to be true for metric spaces, since a metric space is separable if and only if it is Lindelof (see e.g., Engelking Gen Top 1989, 4.1.16)... | https://mathoverflow.net/users/20300 | Does every Lindelof uniform space have a Lindelof completion? | I am entering an "answer" because don't know any other way to mark the question settled. KP Hart added a comment suggesting the space S x S discussed above is a counterexample. This is correct because whenever D is a dense subset of a complete uniform space X, then X is the completion of D in the inherited uniformity. ... | 1 | https://mathoverflow.net/users/20300 | 85264 | 50,787 |
https://mathoverflow.net/questions/84467 | 5 | What is it known now about Nagata's conjecture and Seshadri constant (<http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves> and <http://en.wikipedia.org/wiki/Seshadri_constant>) for toric surfaces? It seems that it should be some lower bounds in terms of fans or polytops. Is it true?
Does there exist some si... | https://mathoverflow.net/users/4298 | Nagata's conjecture, Seshadri constant | I assume $S$ is a projective smooth toric surface.
If $D\_1, \dots, D\_n$ are the irreducible toric divisors on $S$, then $-K\_S=D\_1+\dots+D\_n$ is an anticanonical divisor. Thus blowing up any point on any of these divisors one obtains a smooth anticanonical rational surface $\tilde S$; such surfaces are very well... | 2 | https://mathoverflow.net/users/1939 | 85274 | 50,793 |
https://mathoverflow.net/questions/85282 | 3 | Let's begin with a few observations. Suppose we consider the set of $N\times N$ matrices and consider the matrices with positive determinant. There are several connected components in this set; let $S$ be the component that contains the identity matrix. Observe that $S$ corresponds to the set of positive definite matri... | https://mathoverflow.net/users/8938 | Analogue of PSD matrices for permanents? | The set $C$ is not convex, nor is its intersection with the symmetric matrices. To see this note that by linearly interpolating between each of the matrices below we maintain positive permanent and symmetry:
\[
I = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1\end{bmatrix}
\rightarrow
\begin{bmatrix} 1 & 0 & 0 ... | 6 | https://mathoverflow.net/users/5963 | 85289 | 50,799 |
https://mathoverflow.net/questions/85017 | 2 | Let $X$ be a Tychonof space and $\beta X$ is its compactification. Then could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?
| https://mathoverflow.net/users/18465 | Could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology? | The two sets are essentially the same: the map that sends every $f\in I^{\beta X}$ to its restriction is a bijection; the two topologies are, in general, not the same. The compact-open topology on $I^{\mathbb{N}}$ is the product topology, whereas the compact-open topology on $I^{\beta\mathbb{N}}$ is the topology induce... | 4 | https://mathoverflow.net/users/5903 | 85292 | 50,800 |
https://mathoverflow.net/questions/85298 | 17 | Let $S$ be the (countable) set of holomorphic cuspidal new eigenforms of weight $\geq 2$. Any $f\in S$ has a level $N\_f$ and a canonically normalized Fourier expansion $f(z)=\sum\_{n=1}^{\infty}a\_f(n)e^{2\pi i nz}$ with $a\_f(1)=1$ and $a\_f(n) \in \overline{\mathbf{Z}}$.
Form a graph $\mathcal{G}$ as follows: Tak... | https://mathoverflow.net/users/1464 | The graph of congruences between modular forms | Suppose that $f$ has level $N$, and suppose that $N$ is divisible by $p$.
Then it is well known that $f$ is congruent modulo (some prime above) $p$ to a form
$g$ of level $M$ dividing $N$ (and high weight), where $M$ is prime to $p$.
In particular, by induction, all forms $f$ are connected to a form $g$ of level $1$ in... | 11 | https://mathoverflow.net/users/nan | 85306 | 50,806 |
https://mathoverflow.net/questions/85291 | 3 | Given an extension of groups
$$ 1 \to H \to G \to Q \to 1,$$
there is a spectral sequence
$$E^{ip}\_2(M) = H^i(Q,H^p(H,M)) \Rightarrow H^{i+p}(G,M).$$
I understand that the composition of the cup products for $Q$ and $H$ defines a pairing
$$ E\_2^{ip}(M) \otimes E\_2^{jq}(N)\hspace{180pt}$$
$$\begin{array}{cl}
= & H^... | https://mathoverflow.net/users/18571 | Sign in the product of the LHS spectral sequence | Let $X \to k$ resp. $Y \to k$ be projective resolutions of $k$ over $kG$ resp. $kQ$. In short, the reason for the sign is the twist
$$T : X \otimes Y \to Y \otimes X,\; x \otimes y \mapsto (-1)^{ij} \cdot y \otimes x\quad,\quad x \in X\_i, y \in Y\_j.$$
In detail: First note that if $U \to k$ is a projective resolut... | 4 | https://mathoverflow.net/users/10194 | 85307 | 50,807 |
https://mathoverflow.net/questions/85285 | 5 | Say $f:X\to C$ is a family of curves. More precisely, $C$ is a smooth projective irreducible curve over a field, $f$ is a flat morphism of schemes and $X$ is a normal projective irreducible surface.
Say I take a section $P:C\to X$. Does the image of $P$ lie in the nonsingular part of $X$?
What conditions (weaker th... | https://mathoverflow.net/users/20436 | Does the image of a section lie in the regular part | **Example**:
Let $Y$ be the projective cone over a conic, so for instance, let $X$ be defined by $xz=y^2$ in the projective 3-space with coordinates $[x:y:z:w]$ and consider the projection to the $[x:w]$-axis:
$$ g:Y\dashrightarrow \mathbb P^1$$
$$ [x:y:z:w]\mapsto [x:w]\quad\qquad $$
This is defined everywhere except ... | 10 | https://mathoverflow.net/users/10076 | 85308 | 50,808 |
https://mathoverflow.net/questions/85309 | 12 | Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a constant c (independent of A and B and the dimension) such that
$$(A-B)^2 \leq c (A+B)^2?$$
Thanks.
| https://mathoverflow.net/users/20468 | Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ? | There is no such $c$ even if we use only $2 \times 2$ matrices.
For any $c \geq 1$ let $A,B$ be the positive-semidefinite matrices
$$
A = \left( \begin{array}{lc} c^2 & c \cr c & 1 \end{array} \right),
\phantom\infty
B = \left( \begin{array}{cc} 1 & 0 \cr 0 & 0 \end{array} \right).
$$
of rank $1$. Then we calculate tha... | 29 | https://mathoverflow.net/users/14830 | 85310 | 50,809 |
https://mathoverflow.net/questions/85280 | 3 | I was reading through Akhiezer's book *Lectures on Integral Transforms* and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$,
$f(y) = \int\_0^{\infty}dr \sqrt{ry}J\_{\nu}(ry) \int\_0^{\infty} dx f(x) \sqrt{xr} J\_{\nu}(xr)$.
He leaves the proof tha... | https://mathoverflow.net/users/20460 | Inverse Hankel Transform | <http://en.wikipedia.org/wiki/Hankel_transform#Orthogonality>
This ensures orthogonality. You should think in terms of linear algebra - transition matrix to any orthogonal basis in orthogonal. So you need to prove orhogonality of Bessels. This is stated in Wiki Link above.
PS
Actually I do not quite understand wh... | 1 | https://mathoverflow.net/users/10446 | 85315 | 50,810 |
https://mathoverflow.net/questions/85253 | 2 | Let a convex quadrilateral ABCD with perimeter 1，d is the maximum of AB,AC,AD,BC,BD,CD,prove that d is not less than 1/3
we can prove that parallelogram ABCD with perimeter 1，than one of AC,BD is more than 1/3
but the general case is very difficult to solve.
| https://mathoverflow.net/users/20398 | the minimal diameter of a quadrilateral | The answer given by ε-δ (a kite inscribed in a Reuleaux triangle) can be found in
Ball, D. G. (1973), "A generalisation of π", *Mathematical Gazette* 57 (402): 298–303, doi:[10.2307/3616052](http://dx.doi.org/10.2307%252F3616052), JSTOR [3616052](http://www.jstor.org/stable/3616052);
He doesn't give an explicit pro... | 5 | https://mathoverflow.net/users/440 | 85316 | 50,811 |
https://mathoverflow.net/questions/85313 | 17 | I would like to hear what's known about the homotopy type of smash products of mod-$p^j$ Moore spectra, for $p$ an odd prime.
First, here is what I'm specifically interested in: there is a short exact sequence $$0 \to \mathbb{Z} \xrightarrow{p^j} \mathbb{Z} \to \mathbb{Z}/p^j \to 0.$$ Tensoring this short exact seque... | https://mathoverflow.net/users/1094 | Homotopy type of tensors of Moore spectra | For $p\neq 2$ the answer is yes. It's an easy computation usinghomology decompositions in the sense of Eckmann–Hilton the well known exact sequence
$$\operatorname{Ext}(A,\pi\_{n+1}(X))\hookrightarrow [M(A,n),X]\twoheadrightarrow \operatorname{Hom}(A,\pi\_n(X)),$$
and the computation
$$\pi\_{n+1}(M(A,n))=A\otimes... | 11 | https://mathoverflow.net/users/12166 | 85321 | 50,813 |
https://mathoverflow.net/questions/85324 | 6 |
>
> Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)?
>
>
>
More precisely, let
$$
f = a\_0 + a\_1 X + \cdots + a\_{n-1}X^{n-1} + X^n
$$
with $a\_i \in \mathbb{Z}$ and $\l... | https://mathoverflow.net/users/8153 | bound for zeros of a polynomial with bounded integer coefficients | $\def\conj#1{\overline{#1}}\DeclareMathOperator\Res{Res}$If $z$ is a zero of $f$, then $|z|^2-1=z\conj z-1$ is a zero of the resolvent $g(w)=\Res\_z(\conj f(z),z^nf((w+1)/z))$. You can extract a bound on the (integer) coefficients of $g(w)$ from the definition, and then e.g. Cauchy's bound will give you a lower bound o... | 3 | https://mathoverflow.net/users/12705 | 85327 | 50,815 |
https://mathoverflow.net/questions/85276 | 3 | According to the wikipedia page on "Isoperimetric dimension", the isoperimetric dimension is invariant under quasi-isometries, even between manifolds and graphs:
"[...] the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometrie... | https://mathoverflow.net/users/20458 | Isoperimetric dimension of Graphs. | You can find definitions and properties in Fan Chung's paper,
"Discrete Isoperimetric Inequalities,"
*Surveys in Differential Geometry IX*, International Press, 2004, 53--82
([PDF download link](http://math.ucsd.edu/~fan/wp/iso.pdf)).
She says, "In a way, a graph can be viewed as a
discretization of a Riemannian mani... | 4 | https://mathoverflow.net/users/6094 | 85334 | 50,818 |
https://mathoverflow.net/questions/57202 | 7 | When $\mathfrak g$ is a complex, simple, simply laced Lie algebra of rank r then the (specialized) character of the basic representation for the corresponding affine Lie algebra $\hat {\mathfrak g}$ is given by $\chi\_{\hat {\mathfrak g}}(q)=\frac{\Theta \_{\mathfrak g}(q)}{\eta (q)^r}$, where $\Theta \_{\mathfrak g} (... | https://mathoverflow.net/users/13377 | Character of the Basic Representation for Affine E_8 in Terms of Jacobi Theta Functions | [Your (related?) question](https://mathoverflow.net/questions/74961/getting-certain-modular-functions-from-characters) brought me here. I'm not sure if I'm following you right, but I guess your first question asks: is there {$h\_1,...,h\_8$},a basis of the Cartan subalgebra, such that $${\rm tr}\_{L(\Lambda\_0)} q^{L... | 3 | https://mathoverflow.net/users/nan | 85342 | 50,823 |
https://mathoverflow.net/questions/85352 | 0 | I'm studying the book "Differential forms of algebraic geometry" of Bott, Tu.
At page 75 there is an exercise about the normal bundle of $CP^1$ in $CP^2$, and there is written that the transition function $g\_{01}$ is $Z\_0/Z\_1$.
this is wrong for me! i consider the embedding $[X,Y] \mapsto [X,Y,0]$, saying $u=Z\_0/Z... | https://mathoverflow.net/users/20483 | Normal bundle of $CP^1$ in $CP^2$ | With your notations, the normal bundle is spanned over $\{Z\_1\ne 0\}$ by $\partial/\partial v$. Now, over $\{Z\_0\ne 0\}$, take affine coordinates $x=Z\_1/Z\_0$ and $y=Z\_2/Z\_0$, so that, where defined, you have $x=1/u$ and $y=v/u$. On this chart your normal bundle is spanned by $\partial/\partial y$.
The transition ... | 0 | https://mathoverflow.net/users/9871 | 85355 | 50,829 |
https://mathoverflow.net/questions/85343 | 13 | I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory.
Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book *Gauge theory and variational principles*, or Baez & Muniain's *Gauge fields, knots and gravity*.
But I am more interested something ... | https://mathoverflow.net/users/7519 | References for classical Yang-Mills theory | This is underrepresented in the literature. I have Nakahara and have looked at Frenkel (both listed in other answers) as well as many other "standard" references. The best book reference for classical YM theory that I found was [Rubakov's *Classical Theory of Gauge Fields*.](http://books.google.com/books?id=BxjL6EkIpfU... | 7 | https://mathoverflow.net/users/1847 | 85358 | 50,832 |
https://mathoverflow.net/questions/85346 | 0 | I need a formula for maximum number of hyperedges that a directed hypergraph with *n* vertices can have. Following point is an unresolved issues for me and it keeps coming back to my mind:
* There are different definitions for hyperedges in directed hypergraphs (e.g. some say a hyperedge e = (T(e), H(e)) in which T(e... | https://mathoverflow.net/users/20482 | Maximum number of hyperedges in a directed hypergraph | I doubt there's a completely standard definition.
It seems like this is elementary -- an undirected edge is a size-$k$ subset of $[1,n]$, and a directed edge is an undirected edge together with one of the possible $2^k$ labellings of it with "T" and "H". So the number of different edges is (if, say, empty head- and ... | 0 | https://mathoverflow.net/users/20281 | 85361 | 50,833 |
https://mathoverflow.net/questions/85365 | 0 | Let $S:=k[X\_1,\ldots,X\_n]$ be a polynomial ring over a field $k$, with its natural grading, and let $\mathfrak{p}$ be a *homogeneous* prime ideal of $S$. Also, let $M:=\bigoplus\_{i} M\_i$ be a finitely generated graded $(S/\mathfrak{p})$-module. Write $\mathcal{F}\_P$ for the coherent sheaf on $\mathrm{Proj}\:\:\: S... | https://mathoverflow.net/users/16046 | Generic rank of a coherent sheaf on a projective variety vs. generic rank on the cone | The ranks are the same. Since $M$ is a finitely generated $(S/\mathfrak{p})$-module you can write down a finite presentation of $M$ by twisted (in the sense of twisting the grading) copies of $S/\mathfrak{p}$:
$$ \oplus\_j (S/\mathfrak{p})(-b\_j) \stackrel{\Phi}{\longrightarrow} \oplus\_{i=1}^k (S/\mathfrak{p})(-a\_i... | 2 | https://mathoverflow.net/users/1055 | 85373 | 50,842 |
https://mathoverflow.net/questions/85323 | 24 | For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the category of locally presentable categories (with morphisms being cocontinuous) by assigning to each geometric morphism $(f^\\*, f\_\\*)$ the ... | https://mathoverflow.net/users/1841 | Topos associated to a category | This is described in the paper
>
> Bunge and Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, 1995.
>
>
>
which describes the sense in which the construction you call Spec is analogous to the symmetric algebra construction.
Bunge and Carboni give a biadjunction between the bic... | 22 | https://mathoverflow.net/users/10862 | 85379 | 50,845 |
https://mathoverflow.net/questions/85386 | 7 | Mackey's test for irreducibility of induced representation over $\mathbb{C}$ is:
Let $G$ be a finite group, $H\leq G$, $W$ be a representation of $H$, and $W^x$ be conjugate representation of $H^x=xHx^{-1}$. Then following are equivalent:
(i) $Ind^G\_H(W)$ is irreducible.
(ii) $W$ is irreducible and for each $x\i... | https://mathoverflow.net/users/6761 | Irreducibility of Induced Representation | No. Over $\mathbb R$ let $G$ be the quaternion group of order $8$, $H$ the subgroup of order $2$, $W$ the nontrivial one-dimensional representation.
EDIT For an even simpler example, see Kevin Ventullo's comment!
| 7 | https://mathoverflow.net/users/6666 | 85388 | 50,847 |
https://mathoverflow.net/questions/85376 | 5 | Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$.
Does it follow that that there exists a real polynomial $p$ such that $p$ is non-decreasing on
1. $\;$ $[0,1]$
2. $\;$ all of $\mathbb{R}$
and for all members $x$ of $[0,... | https://mathoverflow.net/users/nan | Stone-Weierstrass for monotone functions | In fact this question is a simple prototype of a serious problem of approximation maintaining additional qualitative properties of a function, with precise error estimates. See
<http://mathworld.wolfram.com/ComonotoneApproximation.html>
for the case of piecewise monotone functions. There are many problems and results... | 8 | https://mathoverflow.net/users/12205 | 85392 | 50,848 |
https://mathoverflow.net/questions/85400 | 5 | Please give suggestions about soft to make symbolic computations with NON-commutative variables.
Typical examples I am interesting - Capelli identities
<http://en.wikipedia.org/wiki/Capelli>'s\_identity
For example let
2x2 matrix X be defined:
$(x\_{11}~~~ x\_{12})$
$(x\_{21}~~~ x\_{22})$
and D is defined:
... | https://mathoverflow.net/users/10446 | Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ? | There is a boat-load of mathematica packages for Lie Algebra computations. Some examples are:
[SuperLie](http://www.equaonline.com/math/SuperLie/SuperLie.pdf)
[Quantum Mathematica.](http://library.wolfram.com/infocenter/MathSource/7622/?affilliate=1)
| 2 | https://mathoverflow.net/users/11142 | 85402 | 50,850 |
https://mathoverflow.net/questions/85387 | 2 | Assume that $I\subset k[x\_1,\ldots,x\_n]$ and $J\subset k[y\_1,\ldots,y\_m]$ are monomial ideals in different rings, and the minimal free resolution of $S/I$ and $S/J$, say $F\_\cdot$ and $G\_\cdot$, are both linear. I believe that $F\otimes G$ is a minimal free resolution for $S/I+J$. Does anyone have any comment for... | https://mathoverflow.net/users/20466 | when tensor complex resolves S/I+J? | There are really two separate things being asked. (1) When is the complex $F\otimes G$ exact? (2) If it is exact, when is $F\otimes G$ a minimal free resolution?
The first question is computed by Tor. Namely $F\otimes G$ is exact if and only if $\text{Tor}\_i(S/I,S/J)=0$ for all $i>0$
I believe that the second ques... | 6 | https://mathoverflow.net/users/4 | 85408 | 50,853 |
https://mathoverflow.net/questions/85407 | 1 | Suppose we have a smooth vector bundle $\pi: E \rightarrow B$ and two sub vector bundles $\pi\_1: E\_1 \rightarrow B\_1$ and $\pi\_2: E\_2 \rightarrow B\_2$ such that
the bases $B\_1$ and $B\_2$ are submanifolds of $B$. Now suppose we would like to intersect both subbundles, that is we would like to define 'something ... | https://mathoverflow.net/users/17267 | Intersection of subvector bundles | For every vector space $V$ we have a difference map
$$ D: V\oplus V\to V,\;\; D(v\_0,v\_1)=v\_1-v\_0$$
whose kernel is the diagonal $\Delta\_V\subset V\oplus V$. More generally, for vector bundles we have a bundle map
$$D: E\oplus E\to E$$
whose kernel is the diagonal sub-bundle $\Delta\_E$. Consider now the re... | 1 | https://mathoverflow.net/users/20302 | 85413 | 50,855 |
https://mathoverflow.net/questions/85399 | 4 | Hello,
i still have a question about positive closed currents. In particular i know that if $X$ is a compact complex manifold and $T$ is a positive closed current of bidegree $(1,1)$ such that
its cohomology class is zero then is itself zero.
Now, is it possible that is trivial, but is still true if the bidegree is gre... | https://mathoverflow.net/users/19637 | Cohomology class of a current | Take any positive $(1,1)$-form $\omega$ on $X$ and let $T$ be a positive $(p,p)$-current. Then, the trace measure
$$
\sigma\_T=\frac{1}{2^{n-p}(n-p)!}T\wedge\omega^{n-p}
$$
is a positive measure on $X$ which dominates the mass measure $||T||$ of $T$. In particular, if $\sigma\_T$ has vanishing total mass then it is z... | 4 | https://mathoverflow.net/users/9871 | 85424 | 50,860 |
https://mathoverflow.net/questions/85419 | 4 | If we are given a simplicial set $X:\Delta^{op} \to Set$, we may regard it as a \emph{simplicial} simplicial set, i.e. a bisimplicial set by composing with the "constant" inclusion $Set \to Set^{\Delta^{op}}$. We may choose to view this as a simplicial diagram of infinity groupoids. Its infinity colimit may be computed... | https://mathoverflow.net/users/4528 | Writing an infinity groupoid as a colimit of sets | For the fat realization and the realization of simplicial spaces (where space here means simplicial set ;) to be weakly equivalent you need the simplicial diagramm to be Reedy cofibrant. This is e.g. the case if all degeneracies are cofibrations which is true here (if I understand your construction correct).
| 5 | https://mathoverflow.net/users/11002 | 85426 | 50,862 |
https://mathoverflow.net/questions/85427 | 5 | Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are classified thanks to Zariski, Mumford, others in this setting and that is not my question. I want to 'run' LMMP of pairs not to... | https://mathoverflow.net/users/1887 | Minimal Model Program for surfaces over algebraically closed fields of characteristic p | In the surface case, MMP in char p is known. See Koll'ar-Kov'ac's preprint on Koll'ar's webpage.
In dimensional 3, the existence of divisorial contractions and flipping contractions is known as EWM (so the target is only known as a algebraic space). See Keel's paper BASEPOINT FREENESS FOR NEF AND BIG LINE BUNDLES.
I... | 8 | https://mathoverflow.net/users/10083 | 85430 | 50,865 |
https://mathoverflow.net/questions/85411 | 7 | This is probably a silly question with which I am stuck however. Let $G$ be a locally compact group. It seems to me that there is a canonical map of $L\_\infty(G)$ into $M=C\_0(G)^{\*\*}$ (the latter is the enveloping von Neumann algebra of $C\_0(G)$). I would reason as follows:
1. Let $I$ be the annihilator of $L\_1... | https://mathoverflow.net/users/19471 | Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? | Since you are in the commutative setting, you can present the construction more simply. $M(G)=L\_1(G)\oplus\_1 S(G)$, where $S(G)$ is the space of complex measures that are singular with respect to Haar measure, and hence $M(G)^∗=L\_\infty (G)\oplus\_\infty S(G)^∗$.
| 6 | https://mathoverflow.net/users/2554 | 85434 | 50,868 |
https://mathoverflow.net/questions/85433 | 2 | Given an undirected graph $G$ and vertices $s, t$, are there any upper bounds on the number of simple paths from $s$ to $t$?
Can these bounds be improved if you know
1) The distance from $s$ to $t$
2) The graph has max degree $\Delta$
3) No two non-adjacent vertices on the path are allowed to be neighbors. For ... | https://mathoverflow.net/users/9896 | Bounds on number of simple paths in graph | If the distance from s to t is 1, or the max degree is 2, then there are at most 2 such paths. Otherwise there are potentially exponentially many such paths even among cubic graphs (think of a cycle of diamonds). Of course, I am considering the extreme case and not looking at forests or other classes of graphs with few... | 3 | https://mathoverflow.net/users/3568 | 85437 | 50,869 |
https://mathoverflow.net/questions/85435 | 1 | Hi all.
I have the following setting: $A, B$ are $\mathbb{Z}$-modules (in my case, $B$ is free and finitely generated) and i have a $\mathbb{Z}$-bilinear map $\phi:A \times B \mapsto \mathbb{Z}$. Now i want to do an "extension" of scalars, meaning that i take an arbitrary commutative ring $R$ with unit ($\mathbb{F}\_... | https://mathoverflow.net/users/20431 | Perfectness of R-Bilinear form preserved under extension of scalars? | As long as your $B$ is free and finitely generated, $A$, being isomorphic to $d\_{\mathbb Z}(B)$, is also free on the same number of generators. Fix free generators $a\_i$ for $A$ and $b\_i$ for $B$. Perfectness of $\phi$ will make the matrix with entries $\phi(a\_i,b\_j)$ have determinant 1. That determinant will rema... | 2 | https://mathoverflow.net/users/6794 | 85438 | 50,870 |
https://mathoverflow.net/questions/85391 | 23 | Looking for an example of a monad that is not strong.
The reason being, a strong monad (wrt cartesian product) is an "applicative functor" (in functional programming); an example of a non-strong monad would be useful to see what's breaking in its "applicativity".
| https://mathoverflow.net/users/20031 | Any example of a non-strong monad? | Here is a class of examples different to Tom's: if your underlying monoidal category C is closed, then a strong monad on C is the same as a C-enriched monad, i.e. one that respects the [enrichment](https://ncatlab.org/nlab/show/enriched+category) of C given by its internal hom (this is why every monad on Set is strong,... | 15 | https://mathoverflow.net/users/4262 | 85449 | 50,876 |
https://mathoverflow.net/questions/85347 | 2 | Assume we have a noncommutative ring $R$ with exactly 2 non-isomorphic simple left modules $S\_1$ and $S\_2$ (up to isomorphism) and an $R$-bimodule $M$, which switches the simples, i.e. $M\otimes\_R S\_1=S\_2$ and $M\otimes\_R S\_2=S\_1$.
Then we have $Hom\_R(S\_i,M\otimes\_R S\_i)=0$ by Schur's lemma ($\*$).
Now ... | https://mathoverflow.net/users/3233 | Are morphisms of finite length modules determined by the behaviour of the simple modules? | The question is too broad in general, but I believe that there is a nice answer for your particular example. The first thing to note is that the given $M$ is a twisted bimodule. Namely, let $\sigma$ be the automorphism of $R$ given by $$\sigma : \begin{pmatrix} r & s \\ xt & u \end{pmatrix} \mapsto \begin{pmatrix} u &t... | 3 | https://mathoverflow.net/users/11791 | 85456 | 50,878 |
https://mathoverflow.net/questions/85251 | 38 | The MathOverflow question [Open source mathematical software](https://mathoverflow.net/questions/19046/open-source-mathematical-software) contains a list of programs that are useful to perform various computational tasks, such as computer algebra systems.
However, evaluating complicated formulas is not all that a pro... | https://mathoverflow.net/users/1898 | Non-computational software useful to mathematicians | This is my short list of math related software not used for computing. I made an effort to list software in descending order with respect to the frequency of use. I left out $\TeX$ and my version control system of choice [CVS](http://savannah.nongnu.org/projects/cvs) since OP was not interested in those.
1. A good e... | 12 | https://mathoverflow.net/users/7442 | 85465 | 50,884 |
https://mathoverflow.net/questions/85468 | 6 | A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and writing the base $n$ representation of the sequence of natural numbers in the reverse order. [Read more](http://en.wiki... | https://mathoverflow.net/users/20174 | Analogue of van der Corput sequence for prime numbers | While $\lbrace v\_p \rbrace$ is clearly not equidistributed in $(0,1)$, it *is* equidistributed in
$$
\Pi\_{10} := [.1,.2) \cup [.3,.4) \cup [.7,.8) \cup [.9,1)
$$
by the prime number theorem (PNT) for arithmetic progressions modulo powers of $10$. In particular, the average tends to $0.55$, the average of the midpoint... | 11 | https://mathoverflow.net/users/14830 | 85470 | 50,885 |
https://mathoverflow.net/questions/59756 | 17 | Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed commutative ring k). Julia Bergner has shown that the category of simplicial categories, i.e. categories enriched over simpl... | https://mathoverflow.net/users/3293 | Model category structure on categories enriched over quasi-coherent sheaves | There were two preprints posted on the arXiv this week that seem to answer the question. They are [Dwyer-Kan homotopy theory of enriched categories](http://arxiv.org/abs/1201.1575) by Fernando Muro and [On the homotopy theory of enriched categories](http://arxiv.org/abs/1201.2134) by Clemens Berger and Ieke Moerdijk.
... | 6 | https://mathoverflow.net/users/3293 | 85489 | 50,897 |
https://mathoverflow.net/questions/85484 | 3 | I'd like to learn about modular forms. My background is mostly computational algebra and group theory, and I've had little-to-no training in complex analysis. I've briefly seen modular forms in a short literature review I did on Monstrous Moonshine. I've been scouting out various books, and most have a reasonably stron... | https://mathoverflow.net/users/16596 | Algebraic approaches to modular forms | You can do a great deal with no analysis whatsoever, by defining modular forms of weight $k$ to be sections of the line bundle $\omega^{\otimes k}$ over the elliptic moduli stack. That sounds quite scary, but it can be made very elementary and concrete after a couple of pages of preparatory discussion. Deligne's "Courb... | 8 | https://mathoverflow.net/users/10366 | 85496 | 50,899 |
https://mathoverflow.net/questions/85490 | 12 | Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F\_x(0,0) = F\_y(0,0) = F\_{xy}(0,0) = 0$ and that $F\_{yy} \not = 0$ and $F\_{xx}(0,0)/F\_{yy}(0,0) < 0$.
Obviously I cannot apply the implicit function th... | https://mathoverflow.net/users/20528 | Implicit function theorem at a singular point? | This is a job for the Morse lemma. The second degree Taylor polynomial of $F(x,y)$ has the form $ax^2+by^2$ where $ab<0$. (You said $>0$ but that can't be what you meant.) The Morse Lemma says, for a sufficiently smooth function of several variables, that if it has zero constant and linear parts and a nondegenerate qua... | 15 | https://mathoverflow.net/users/6666 | 85498 | 50,900 |
https://mathoverflow.net/questions/76122 | 14 | Let $A\in\mathbb{R}^{n\times n}$ and suppose that $A$ is of rank $m\leq n$. Moreover suppose we know $u\_1,\ldots, u\_m \in\mathbb{R}^{n\times 1}$ and $v\_1,\ldots, v\_m \in\mathbb{R}^{n\times 1}$ such that
$
A = \sum\_{k=1}^m u\_kv\_k^T
$
Is there a faster way than $\mathcal{O}(n^2)$ for finding the minimum (or m... | https://mathoverflow.net/users/2011 | Finding minimum (or maximum) element of a low rank matrix. | Here's a geometrical reformulation of the problem that yields an
$O(n \log n)$ solution for $m=2$ and suggests a context that
may yield good answers for arbitrary fixed $m$.
[**EDIT** but see below for $m=3$ and $m \geq 4$.]
Denote the $i$-th coordinate of $u\_k^{\phantom.}$ and $v\_k^{\phantom.}$ by
$u\_k^{(i)}$ and... | 5 | https://mathoverflow.net/users/14830 | 85501 | 50,901 |
https://mathoverflow.net/questions/85495 | 1 | I'm reading Proposition 14, page 15, Chapter I, taken in Lang - Algebraic number theory. It states that if $A$ is an integrally closed domain with field of fractions $K$, $L$ be a finite galois extension of $K$, $B$ be the integral closure of $A$ in $L$, $\mathfrak P$ be maximal ideal of $B$ lying over a maximal ideal ... | https://mathoverflow.net/users/20529 | A proof in Lang - Algebraic number theory | Lang actually proves that for every $\overline{x}\in \overline B$ (no separability condition required), there exists a polynomial over $\overline A$ which has $\overline{x}$ as a root and splits into linear factors over $\overline B$. This yields that $\overline B$ is normal over $\overline A$.
| 3 | https://mathoverflow.net/users/2530 | 85502 | 50,902 |
https://mathoverflow.net/questions/85505 | 9 | Main Question
-------------
Let $R$ be a graded ring, graded by the nonnegative integers. Denote by $\mathrm{gr}R-\mathrm{Mod}$ the category of $\mathbb{Z}$-graded left $R$-modules with morphisms that preserve the grading. Is there a ring $S$ such that the category $S-\mathrm{Mod}$ of left $S$-modules is equivalent t... | https://mathoverflow.net/users/703 | Module category equivalent to graded module category? | $\newcommand\ZZ{\mathbb{Z}}$Let $R=\bigoplus\_{n\in\mathbb N\_0}R\_n$ be your graded ring. Construct a category $Q$ with objects $\ZZ$ and where $\hom\_Q(n,m)=R\_{m-n}$ with composition coming from the multiplication in $R$. A graded left $R$-module is the same thing as a functor from $Q$ to abelian groups. The (non-un... | 7 | https://mathoverflow.net/users/1409 | 85515 | 50,906 |
https://mathoverflow.net/questions/85246 | 6 | Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi\_{\mathscr{X}}:\int\_{C} \mathscr{X}\to C$$ denote the associated fibered category. The underlying category $\int\_{C} \mathscr{X}$ carries an induced Grothendieck topology such that $$St\left(... | https://mathoverflow.net/users/4528 | Why are sheaves not preserved in this case? | Here's what's wrong:
The inclusion $j:Sh(C) \to St(C)$ does not preserve colimits. Notice that $j$ has a right-adjoint given by $\pi\_0,$ at least making sheaves reflective. To see that $j$ does not preserve colimits, take for instance the colimit of $\pi\_\mathscr{X}$, first by composing with the Yoneda embedding in... | 1 | https://mathoverflow.net/users/4528 | 85524 | 50,911 |
https://mathoverflow.net/questions/85527 | 11 | Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int x\mu(dx)=1/2\right)$$
This is a convex set, with convex combinations defined as $\mu=\alpha \eta +(1-\alpha)\xi$ when $... | https://mathoverflow.net/users/18474 | Extreme points of a set of probability measures | This question (where you prescribe a set of moments, on an arbitrary measure space) is completely answered in [this very cool paper.](http://dl.dropbox.com/u/5188175/971911.pdf) (G. Winkler, Extremal points of moment sets).
| 14 | https://mathoverflow.net/users/11142 | 85530 | 50,916 |
https://mathoverflow.net/questions/85539 | 3 | **Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$.**
**Is there a way to know if this is *always* a non-positive (sectional) curvature manifold?**
Note this is a parametrized manifold that is locally conformally flat.
Following *Einstein Manifolds* [Arthur L. ... | https://mathoverflow.net/users/20458 | Conformally-flat | I'm not quite sure what you mean by *always* non-positively curved. If you are asking if this metric is non-positively curved for any $f$ then this is false. If you are asking for conditions on $f$ ensuring that the resulting metric is non-positively curved then there is a general formula:
Let $(M,g)$ be a Riemannian... | 10 | https://mathoverflow.net/users/18050 | 85542 | 50,922 |
https://mathoverflow.net/questions/85540 | 23 | What are the groups $X$ for which there exists a group $G$ such that $G' \cong X$?
My considerations:
---
$\bullet$ If $X$ is perfect we are happy with $G=X$.
$\bullet$ If $X$ is abelian then $G := X \wr C\_2$ verifies $G'=\{(x,x^{-1}): x \in X\} \cong X$.
$\bullet$ If $X$ satisfies the following properties... | https://mathoverflow.net/users/5710 | Realizing groups as commutator subgroups | A complete answer seems not to be known. Let me give you the following two nearly-contemporaneous references from the mid-70s:
Robert Guralnick, *On groups with decomposable commutator subgroups*
Michael Miller, *Existence of Finite Groups with Classical Commutator Subgroup*
Both Guralnick and Miller call groups ... | 24 | https://mathoverflow.net/users/35575 | 85545 | 50,923 |
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