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/35910 | 4 | A k-coloring or k-labeling of the vertices of a single-component undirected graph G with $n$ vertices can be a proper coloring or not. If it is not a proper coloring, such that each vertex has neighbors on its edges which are of a different color or label, then for each possible labeling, it possible to count the numbe... | https://mathoverflow.net/users/8567 | Tractably Partitioning the possible vertex k-colorings of a graph by local stability and instability. | If you limit it to specific classes of graphs, say for example star graphs, you can come up with some answers. For a star graph $S\_m$, with a vertex at the center and $m$ vertices connected to the center, yielding a graph $G$ with $n=m+1$ vertices and $m$ edges, it can be calculated that for $k=2$
If the center vert... | 3 | https://mathoverflow.net/users/8636 | 36115 | 23,269 |
https://mathoverflow.net/questions/36088 | 1 | What are the functions $f$ so that a set $\{a \cdot f(x+b) : a \in \mathbb{R}, b \in \mathbb{R}\}$ is a finite dimensional linear vector space ?
Is there a complete characterization of such functions?
$e^{c x}$ (where $c$ is some constant) is a good example of a base of one dimensional space.
$sin (c x), cos(c x... | https://mathoverflow.net/users/8631 | Linear space of translatable functions. | The question, as stated, is about the *set* of multiples of translates, but from the example quoted, $\sin x,$ I suspect that OP really meant the *span*.
**Theorem** Let $f$ be a continuous complex-valued function on $\mathbb{R}.$ Then the following conditions are equivalent:
1. The translates $\{f(x+b) : b\in\mat... | 6 | https://mathoverflow.net/users/5740 | 36119 | 23,272 |
https://mathoverflow.net/questions/36091 | 8 | If we consider the action of the $U\_p$ operator on overconvergent $p$-adic modular forms, then we can get some information about the field over which the eigenforms are defined by looking at the slopes. For instance, my paper in Math Research Letters ([MR2106238](http://www.ams.org/mathscinet/search/publdoc.html?arg3=... | https://mathoverflow.net/users/4555 | Fields of definition for p-adic overconvergent modular eigenforms | Professor Buzzard raises the a question of whether *every* normalized eigenform of level $1$ is defined over a quadratic extension of $\mathbb{Q}\_2$.
(This is Question 4.3 of <http://www2.imperial.ac.uk/~buzzard/maths/research/papers/conjs.pdf>)
In contrast, the multiplicity of the valuation of the set of $2$-adi... | 2 | https://mathoverflow.net/users/nan | 36120 | 23,273 |
https://mathoverflow.net/questions/36126 | 26 | Next semester I will be teaching an introductory course on geometric group theory and there is a basic question that I do not know the answer to. Let $G$ be a finitely generated group with finite symmetric generating set $S$ and let $\Gamma$ be the corresponding Cayley graph. For each $n \geq 1$, let $B\_{n}$ be the cl... | https://mathoverflow.net/users/4706 | On the size of balls in Cayley graphs | In the article
[R. Grigorchuk and P. De La Harpe, On problems related to growth, entropy, and spectrum in group theory, Journal of Dynamical and Control Systems, Volume 3, Number 1, 51-89](http://www.springerlink.com/content/475575746034r344/)
on the lower part of page 58 the authors mention the manuscript
A. Ma... | 31 | https://mathoverflow.net/users/8176 | 36132 | 23,279 |
https://mathoverflow.net/questions/29256 | 6 | I have a vertex set $V$ and a collection of disjoint arc sets $E\_1, \ldots, E\_t$ such that $$G\_i = (V, E\_i),\quad\forall i = 1, \ldots t,$$ are directed acyclic graphs (DAGs) and $$G = (V, E\_1 \cup \ldots \cup E\_t)$$ is a tournament. We note that the individual DAGs may be disconnected and that $G$ may not be acy... | https://mathoverflow.net/users/685 | Combining DAGs into an acyclic tournament | The problem you pose, of finding a bipartition if one exists, is of polynomially equivalent difficulty to the decision problem of determining whether a bipartition exists. The decision problem in turn is NP-complete, by reduction from 3-SAT (and the fact that a solution is easily checked.)
Given an instance of 3-SAT ... | 2 | https://mathoverflow.net/users/7936 | 36137 | 23,283 |
https://mathoverflow.net/questions/36084 | 0 | Hello to all,
While sprucing up my knowledge of group (co)homology,I stumbled onto the following question: The first step you usually take to compute various (co)homologies is to construct the infamous "bar resolution" which resolves $\mathbb{Z}$ by free $\mathbb{Z}[G]$-modules (I'll assume everyone knows which one I... | https://mathoverflow.net/users/4863 | (co)homology of cyclic groups |
>
> Louis asked: "I was wondering if it was possible to distill this 2-periodic resolution somehow out of the standard bar-resolution above in some natural way ?"
>
>
>
By a result of Benson and Carlson [Complexity and Multiple Complexes. Math. Zeit. 195(1987), 221-238, Theorem 4.4], for finite groups there is ... | 4 | https://mathoverflow.net/users/8644 | 36144 | 23,286 |
https://mathoverflow.net/questions/36154 | 1 | I have an intuition that Average of twin prime pairs is always Abundant number except for 4 and 6.
For example:
12 < 1+2+3+4+6=16
18 < 1+2+3+6+9=21
...
But I can't prove this. Could you give me any good idea?
---
2010-08-22
I think that Any prime is a factor of average of twin prime pair.
Do you agree w... | https://mathoverflow.net/users/8140 | Average of twin prime pairs is Abundant number, except for 4 and 6. Any prime is a factor of average of twin prime pair | Prove the sum is always a multiple of 6, then prove that multiples of 6 are abundant.
| 9 | https://mathoverflow.net/users/3684 | 36155 | 23,291 |
https://mathoverflow.net/questions/36169 | 6 | Hi all,
I would like to propose the following problem:
Given two representations $\rho$ and $\tau$ of a group $G$ over complex number, we would like to know if there exists an automorphism $\phi$, such that $\rho\circ\phi$ and $\tau$ are equivalent.
Is there any mathematical results concerning this problem? It s... | https://mathoverflow.net/users/8012 | to test equivalence of representations under automorphism | I assume that $\phi$ is an automorphism of $G.$ Note that if $\phi$ is inner then trivially $\rho$ and $\rho\circ\phi$ are equivalent, thus the answer depends only on the image of $\phi$ in the outer automorphism group $Out(G).$
If $G$ is a finite group (or, more generally, compact group) and representations are fin... | 13 | https://mathoverflow.net/users/5740 | 36170 | 23,296 |
https://mathoverflow.net/questions/36175 | 14 | In the elementary group theory we know that for the symmetric groups $S\_n$, except $S\_6$, we have $Aut(S\_n) \cong S\_n$. Then the following question is natural:
What is the necessary and sufficient condition for $G$ such that $Aut(G) \cong G$?
| https://mathoverflow.net/users/3926 | Is there any criteria for whether the automorphism group of G is homomorphic to G itself? | This answer is essentially a series of remarks, but ones which I hope will be helpful to you.
(1) There are two ways to interpret the condition that $G$ be isomorphic to its automorphism group: canonically and non-canonically.
a) Say that $G$ is [complete](http://en.wikipedia.org/wiki/Complete_group) if every autom... | 14 | https://mathoverflow.net/users/1149 | 36177 | 23,299 |
https://mathoverflow.net/questions/36128 | 9 | Let $q$ and $q'$ be complex numbers with $0<|q|,|q'|<1$, and let $m$ and $n$ be positive integers.
Suppose that $q^m={q'}^n$. Then the map
$$
f:\mathbb{C}^\times/q^{\mathbb{Z}} \to \mathbb{C}^\times/{q'}^{\mathbb{Z}}\qquad \text{defined by}\qquad u\mapsto u^m
$$
gives an isogeny of (analytic) elliptic curves over $\... | https://mathoverflow.net/users/437 | Isogenies between Tate curves | No matter how you define Tate(q), it should have the following properties:
(a) for any $n$ it contains a subgroup $M\_n$ canonically isomorphic to $\mu\_n$ (which corresponds tho $\mu\_n\subset\mathbb{C}^\times$ in the complex model),
(b) the (co)tangent space along the unit section is canonically trivialized (by $... | 7 | https://mathoverflow.net/users/7666 | 36179 | 23,300 |
https://mathoverflow.net/questions/36178 | 13 | Warning: This is a very stupid question regarding a basic misunderstanding that I have. I realize that the question is very elementary, but I guess asking stupid questions is better than remaining ignorant.
To be explicit, consider the $\mathrm{SU}(2)$ Chern-Simons action on some very nice $3$-manifold $M$, i.e. the ... | https://mathoverflow.net/users/8653 | What is the trace in the Chern-Simons action? | The trace is simply a (properly normalised) ad-invariant inner product on the Lie algebra; that is, a nondegenerate symmetric bilinear form $\langle-,-\rangle$ which obeys the "associativity" condition
$$\langle [x,y],z \rangle = \langle x, [y,z] \rangle$$
for every $x,y,z$ in $\mathfrak{g}$.
Lie algebras admitting s... | 19 | https://mathoverflow.net/users/394 | 36180 | 23,301 |
https://mathoverflow.net/questions/36156 | 2 | I'm working out of *Sheaves in geometry and logic*, for reference.
There is a characterisation of flat functors $A:C \to Set$ as those such that the Grothendieck construction $\int\_C A$ is a filtering category. There are more general versions of this result, in which $Set$ is replaced by a more general topos. One sh... | https://mathoverflow.net/users/4177 | internal version of a flat functor? | A flat internal presheaf/discrete fibration $F \to C$ is simply one whose total category F is filtered in the internal sense (see *Topos Theory* 2.51 (filteredness), 4.31 (flatness); *Elephant* B.2.6.2, B.3.2.3). The definition just reexpresses filteredness by requiring that certain maps (e.g. $F\_0 \to 1$) are regular... | 5 | https://mathoverflow.net/users/4262 | 36182 | 23,303 |
https://mathoverflow.net/questions/36190 | 1 | I have the following question about short-time existence and uniqueness
results for non-linear schrodinger equations (NLSE) where the non-linearity involves
a loss of derivatives (in my case, it is a non-local non-linearity involving
a loss of two derivatives.) It seems that most current techniques allow
some small nu... | https://mathoverflow.net/users/6223 | Short-time Existence/Uniqueness for Non-linear Schrodinger with Loss of Several Derivatives | Just a few thoughts. The answer to your question depends on a number of key factors. To focus, let us consider a nonlinear term like $F(D\_x^k u)$ and let us work in one space dimension.
1) How large is $k$? If $k\le2$ you can linearize the equation and work in Sobolev spaces. Of course you need some structural assum... | 5 | https://mathoverflow.net/users/7294 | 36196 | 23,308 |
https://mathoverflow.net/questions/36183 | 13 | This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model.
I gauge from some other interactions that it has considerable interest even in mathematics (geometry?) apart from QFT.
I would like if people can type in what are the canonical e... | https://mathoverflow.net/users/2678 | Linear/Non-linear sigma model | I don't know anything about the QFT side, so I'll refrain from saying things about it.
For the mathematics, one of the reasons that there aren't that many expository/introductory references for it maybe because the development of the (non-linear) theory is rather incomplete. (The linear theory is sort-of trivial: it... | 8 | https://mathoverflow.net/users/3948 | 36198 | 23,310 |
https://mathoverflow.net/questions/36140 | -1 | I've seen some definition of the relative entropy between two states of a C\*algebra. However this definitions work only for finite dimensional C\*algebras and I don't know if there is a correspondent notion for only one state.
| https://mathoverflow.net/users/8642 | Is there a general notion of entropy for the states of a C*algebra? | There are several different definitions of entropy associated with operator algebras. It would be good if you knew which one you were referring to. Do you have a reference? I expect you're talking about the generalization of Kullback-Leibler relative entropy to matrix algebras.
There's a perfectly good definition of... | 1 | https://mathoverflow.net/users/2294 | 36200 | 23,311 |
https://mathoverflow.net/questions/33675 | 2 | The basis for the deterministic polynomial-time algorithm for primality of Agrawal, Kayal and Saxena is (the degree one version of) the following generalization of Fermat's theorem.
---
Theorem
-------
Suppose that P is a polynomial with integer coefficients, and that p is a prime number. Then
$(P(X))^p\equiv P... | https://mathoverflow.net/users/3651 | Fermat for polynomials, as used in the AKS (Agrawal-Kayal-Saxena) algorithm | Your first theorem occurs as an easily proved statement on p. 287 of Schönemann's article *Grundzüge einer allgemeinen Theorie der höhern Congruenzen, deren Modul eine reelle Primzahl ist*, J. Reine Angew. Math. 31 (1846), 269--325. Schönemann was one of the first mathematicians (not counting Gauss, who eliminated the ... | 5 | https://mathoverflow.net/users/3503 | 36202 | 23,313 |
https://mathoverflow.net/questions/36199 | 7 | What kinds of Yoneda-like situations induce an embedding that preserves the tensor product for some arbitrary monoidal category?
The cases where the monoidal product is given by a limit or colimit give this immediately for the usual Yoneda embedding, but this breaks down for "real" monoidal categories like $(Vect, \o... | https://mathoverflow.net/users/800 | In what cases does a Yoneda-like embedding preserve monoidal structure? | Day showed that, for suitable V, any monoidal structure on a (V-)functor category $[C^{\mathrm{op}}, V]$ is essentially determined by its restriction to the representables as
$$ F \otimes G = \int^{A,B} F A \otimes G B \otimes P(A,B,-) $$
where $P(A,B,-) = C(-, A) \otimes C(-, B)$ is a profunctor $C \otimes C \otimes C... | 5 | https://mathoverflow.net/users/4262 | 36203 | 23,314 |
https://mathoverflow.net/questions/36208 | 1 | Let $A$ be a commutative euclidean ring, (probably) with 2 a unit in $A$. I'm trying to compute Witt and Grothendieck-Witt rings and since $A$ is a PID any f.g.p. module over it is free so I only need think about forms on $A^n$.
Question: Is every (non-degenerate) quadratic form over $A$ diagonalizable?
A form $q$... | https://mathoverflow.net/users/1123 | Diagonalization of quadratic forms over euclidean rings | Quadratic forms over $\mathbb{Z}$ don't diagonalize in general.
Even positive definite rank two forms like $3x^2+2xy+5y^2$ can't be diagonalized.
Inverting $2$ won't help things.
| 3 | https://mathoverflow.net/users/4213 | 36210 | 23,318 |
https://mathoverflow.net/questions/36050 | 9 | This is a follow up question to my answer here [How do you define the Euler Characteristic of a scheme?](https://mathoverflow.net/questions/35156/how-do-you-define-the-euler-characteristic-of-a-scheme/36038#36038)
A real analytic space is a ringed space locally isomorphic to $(X,O/I)$ where $X$ is the zero locus of s... | https://mathoverflow.net/users/2349 | Embeddings and triangulations of real analytic varieties | If you just want a proper 1-1 real analytic map whose image is a real analytic variety then the result is theorem 2 page 593 of a paper of Tognoli and Tomassini in Ann.Scuola.Norm.Pisa
(3) vol 21 yr 1967 pages 575-598. This means there is no control over the differential of the
map.I am assuming that the real analytic ... | 5 | https://mathoverflow.net/users/4696 | 36213 | 23,319 |
https://mathoverflow.net/questions/36222 | 32 | I just discovered that something I've been working with has the structure of an operad. So I'm wondering what natural basic questions does one ask about operads? For example, if I knew I had the structure of a group, I'd ask if it is abelian or has torsion, etc. So what are these questions for operads?
| https://mathoverflow.net/users/8667 | What are natural questions to ask about an operad? | There's a lot of things you could ask.
* Operads act on things, that's their point. What things does your operad act on? Presumably this is how you found your operad. Moreover, once you know it acts on something you can ask if that action is maximal, whether or not your operad fits into a bigger operad that also act... | 19 | https://mathoverflow.net/users/1465 | 36223 | 23,322 |
https://mathoverflow.net/questions/36225 | 1 | Suppose *f* is a real-valued function of one variable, and suppose *f* is of differentiability class C1. My question is, if $\Gamma$ is the graph of *f*, then must $\dim\_H(\Gamma)=1$? If anyone knows of a published proof of the answer, I'd appreciate a reference greatly.
| https://mathoverflow.net/users/8665 | Do all graphs of C1 functions have Hausdorff dimension 1? | Let $f \colon I \to \mathbb{R}$. Since $f$ is $C^1$, the graph $\Gamma\_f$ is locally bilipschitz to $I$, via the projection. It follows that Hausdorff dimension is the same as that of $I$ (being defined in terms of the metric space structure only), so it is $1$.
Disclaimer: I haven't seen these topics for quite a wh... | 2 | https://mathoverflow.net/users/828 | 36233 | 23,325 |
https://mathoverflow.net/questions/36218 | 9 | It is well known that the automorphisms of a group $G$ form a group under composition, and that the group of inner automorphisms $\phi (x)=gxg^{-1}$ forms a normal subgroup of $\mbox{Aut}(G)$. Thus, $\mbox{Aut}(G)$ is simple if and only if either $\mbox{Inn}(G)=\mbox{Aut}(G)$ or $\mbox{Inn}(G)$ is trivial. In the secon... | https://mathoverflow.net/users/6856 | Criteria for Aut(G) to be simple | Here is an approximation of an answer to "For what finite groups is Aut(G) simple?"
As Daniel Miller mentioned, Inn(G) is a normal subgroup of Aut(G), so for Aut(G) to be simple either Inn(G) = 1, in which case G is abelian, or Inn(G) = Aut(G) is simple. The former case should be somewhat easy to handle assuming G is... | 21 | https://mathoverflow.net/users/3710 | 36239 | 23,331 |
https://mathoverflow.net/questions/33125 | 2 | Consider a fourth order linear (biharmonic) PDE in two variables of the form
$\nabla^4u + c\nabla^2u-\lambda u = F(x,y)$; $(x,y) \in \Lambda$
To have uniqueness, we must specify two equations per point on $\partial \Lambda$. Now consider the limit where $c\rightarrow \infty$. The solution, $u$, is approximated by $... | https://mathoverflow.net/users/5312 | What happens to the boundary conditions as a PDE is approximated by a lesser order PDE? | The question seems to contain a misprint, as Harald Hanche-Olsen pointed out above. I am assuming that the real situation is as follows: $ \epsilon\nabla^4u + c\nabla^2u-\lambda u=F(x,y)$, where $ (x,y)\in\Gamma $, with a pair of boundary conditions, say $ u=u'=0$, where $ (x,y)\in\delta\Gamma $.
When $\epsilon$ is "... | 3 | https://mathoverflow.net/users/8670 | 36246 | 23,335 |
https://mathoverflow.net/questions/36251 | 10 | I'm looking into a secret sharing scheme that has a secret permutation $\theta$ which has the cycle structure (n/2)+(n/2) (i.e. two (n/2)-cycles).
The permutation $\theta$ is decomposed into two permutations $\alpha$ and $\beta$, where $\alpha$ is generated uniformly at random. So with knowledge of both $\alpha$ and ... | https://mathoverflow.net/users/2264 | What can I say about the permutation $\alpha\beta$ if I know the permutation $\beta\alpha$? | $\theta$ could be any permutation of the form $\alpha (\beta \alpha) \alpha^{-1}$; in other words, it could be any permutation conjugate to $\beta \alpha$, so knowing $\beta \alpha$ tells you only the cycle type of $\theta$, no more and no less. Since you already specified the cycle type this means an attacker gains no... | 25 | https://mathoverflow.net/users/290 | 36252 | 23,338 |
https://mathoverflow.net/questions/31116 | 3 | Hello,
Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also mention some papers if you know.
Thank you !
| https://mathoverflow.net/users/6953 | Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry | Lubotzky and Zuk's book [on property ($\tau$)](http://www.ma.huji.ac.il/~alexlub/) discusses expander graphs and the minimal eigenvalue of the Laplacian on covers of Riemann surfaces. See for example Prop. 2.9 in the book. There's also his book D[iscrete groups, expanding graphs and invariant measures](http://books.goo... | 4 | https://mathoverflow.net/users/1345 | 36253 | 23,339 |
https://mathoverflow.net/questions/35396 | 5 | Let $F\_n$ where $n \ge 3$ be a free group and let $(\mathcal A\_n(k))$ where $k \ge 1$ be the
kernel of the homomorphism $Aut(F\_n) \to Aut(F\_n/\gamma\_{k+1}(F))$
determined by the natural homomorphism $F\_n \to F\_n/\gamma\_{k+1}(F).$
($(\mathcal A\_n(k) : k \ge 1)$ is called the Johnson filtration of $Aut(F\_n);$... | https://mathoverflow.net/users/7983 | Can all terms of the Johnson filtration be hom-mapped onto the same nontrival group? | The answer seems to be affirmative. We use the idea of Henry Wilton that the image might be taken as an alternating group $A\_q$, a *simple* one (see his comment above). Let $K=\mathcal A(1).$ Then
$\mathcal A(m) \ge [K,K,\ldots,K]=[..[K,K],..,K]\qquad (m\quad times) \qquad (\*)$
Take a nontrivial $\alpha \in [K,K]... | 5 | https://mathoverflow.net/users/7983 | 36258 | 23,342 |
https://mathoverflow.net/questions/36263 | -3 | There are three boxes. B1, B2, B3 The probability of selecting them is 0.2, 0.2 , 0.6 respectively.
B1 contains 3 red balls and 7 green balls.
B2 contains 5 red balls and 5 green balls.
B3 contains 2 red balls and 8 green balls.
If we select a box and then a ball from the box what is the probability that the ball ... | https://mathoverflow.net/users/8246 | Porbability of selecting balls from boxes | Well, this looks more like someone trying to get their homework done, but for the first part:
$ p = 0.2 \* \frac{3}{3+7} + 0.2 \* \frac{5}{5+5} + 0.6 \* \frac{2}{2+8}$
$ p = 0.06 + 0.10 + 0.12 $
$ p = 0.28 $
Showed the work for you too.
So if the probability that a chosen ball is red is 28%, then the probabil... | 0 | https://mathoverflow.net/users/8676 | 36265 | 23,345 |
https://mathoverflow.net/questions/36276 | 14 | Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary.
As the question suggests, I would like to know when $\sin(p\pi/q)$ can be expressed in radicals (in the way that $\sin(\pi/4) = \sqrt{2}/2$ and $\sin(\pi/3) = \sqrt{3}/2$ can). L... | https://mathoverflow.net/users/4362 | When is sin(r \pi) expressible in radicals for r rational? | As $\cos x=\pm\sqrt{1-\sin^2 x}$ and $e^{ix}=\cos x +i\sin x$, and
$\sin x=(e^{ix}-1/e^{ix})/2i$ then $\sin x$ is in a radical extension of $\mathbb{Q}$
iff $e^{ix}$ is. For rational $r$ with denominator $d$, $e^{2\pi i r}$
is a primitive $d$-th roots of unity. The extension of $\mathbb{Q}$
generated by a root of unity... | 29 | https://mathoverflow.net/users/4213 | 36277 | 23,352 |
https://mathoverflow.net/questions/36278 | 6 | Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $I\subset G$ the inertia group and $P$ the maximal pro-$p$ subgroup of $I$. Let $I\_t:=I/P$.
Let $A\_0$ be an abelian scheme over $... | https://mathoverflow.net/users/8680 | Abelian varieties over local fields | $V$ is an irreducible $\mathbb{F}\_p$-representation of $I$. As $P$ is a pro-$p$ group, $V^P\ne 0$, and $P$ is normal in $I$ so $V^P$ is stable under $I$. Therefore $V^P=V$. You might like to look at Tate's article on finite flat group schemes in "Modular Forms and Fermat's Last Theorem" for more information.
| 7 | https://mathoverflow.net/users/5480 | 36280 | 23,353 |
https://mathoverflow.net/questions/36281 | 6 | Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree ([sequential spaces](http://en.wikipedia.org/wiki/Sequential_space)). Of course we all know that metric spaces and more gener... | https://mathoverflow.net/users/3041 | Sequential topological vector spaces | The space of tempered distributions is sequential (for its usual strong topology). See, e.g., [Dudley](http://www.ams.org/journals/proc/1971-027-03/S0002-9939-1971-0270145-X/S0002-9939-1971-0270145-X.pdf), and the references therein.
| 12 | https://mathoverflow.net/users/2508 | 36300 | 23,362 |
https://mathoverflow.net/questions/36309 | 0 | What is the number of lines that pass through the center of an n-dimensional tic-tac-toe grid?
| https://mathoverflow.net/users/8682 | tic-tac-toe n-dimensional | The squares tic tac toe grid can be represented with $(x^1,x^2,...,x^n)$ where each can be 0, 1 or 2.
A line passing through the center can be identified Given any coordinate except $(0,0,...,0)$.
There are $3^n-1$ possible coordinates to start a line from.
For each line through center, there will be two endpoint... | 2 | https://mathoverflow.net/users/8602 | 36310 | 23,369 |
https://mathoverflow.net/questions/36105 | 9 | I recall hearing about a result, or maybe a cluster of results, in some area of complexity theory, probably algebraic, to the effect that there are known, specific, short formulas whose minimal derivation is known to be exceedingly long. Or perhaps it is a specific function that requires an exceedingly deep curcuit. Th... | https://mathoverflow.net/users/1643 | nonasymptotic complexity results | Perhaps you were told about
>
> Larry J. Stockmeyer, Albert R. Meyer: Cosmological lower bound on the circuit complexity of a small problem in logic. J. ACM 49(6): 753-784 (2002)
>
>
>
From the abstract: *"An exponential lower bound on the circuit complexity of deciding the weak monadic second-order theory of ... | 4 | https://mathoverflow.net/users/2618 | 36313 | 23,371 |
https://mathoverflow.net/questions/36286 | 5 | Is every holomorphic vector bundle a direct summand of a trivial vector bundle on submanifolds of C^n? What about projective varities? I believe Swan's theorem says something about the first question. But I wanted to make sure.
| https://mathoverflow.net/users/3709 | Holomorphic vector bundles and Swan's theorem | The statement for Stein manifolds follows indeed from the analogue of the Serre-Swan theorem for Stein manifolds, which was proven first in 1967 in "Zur Theorie der Steinschen Algebren un Moduln" by O. Forster. The situation is a bit more complicated than the affine scheme or manifold case, but the final result relevan... | 8 | https://mathoverflow.net/users/6986 | 36315 | 23,372 |
https://mathoverflow.net/questions/36230 | 3 | Let $F\colon C\to D$ be a functor. Given a functor $\delta\colon D\to Sets$, one can compose with $F$ to get $\delta\circ F\colon C\to Sets$. This process is functorial in the category of $D$-sets, and we denote it $$F^\ast\colon D-set\to C-set.$$ The functor $F^\ast$ has both a left and a right adjoint, denoted $F\_!$... | https://mathoverflow.net/users/2811 | Does F_* have a description in terms of the Grothendieck construction? | I guess it depends what you would accept as a "description in terms of the Grothendieck construction."
For each $d \in D$, we have the projection $\pi\_d : d/F \rightarrow C$. Given some $\gamma : C \rightarrow Set$, I can form the composite $$ \gamma \circ \pi\_d : d/F \rightarrow C \rightarrow Set$$
This determi... | 2 | https://mathoverflow.net/users/4466 | 36317 | 23,374 |
https://mathoverflow.net/questions/36312 | 4 | Consider a Seifert fiber space. Is it always possible to find a finite cover that is a circle bundle and the preimage of any fiber is a finite union of circles?
| https://mathoverflow.net/users/3375 | Getting rid of exceptional fibers by passing to finite covers? | If the Seifert fiber space is compact, then this is true, as long as the base orbifold is "good", which means that it has a finite-sheeted manifold cover, which is a compact surface. This induces a cover of the Seifert fiber space which is a circle bundle over the surface. If the base orbifold is bad, then no such cove... | 6 | https://mathoverflow.net/users/1345 | 36323 | 23,378 |
https://mathoverflow.net/questions/36326 | 33 | I'm finishing up a PhD in math and am thinking about options outside of academia. So far, I've really only focused on pure mathematics, but I have a year left of grad school. Suppose I am interested in looking for a job in software engineering, finance, or some other quantitative field. What should I be doing in the ne... | https://mathoverflow.net/users/8694 | How to transition from pure math PhD to nonacademic career? | This is the perspective of someone who went from a math PhD to the media industry and then to software/tech (and enjoyed it immensely). I can't speak for finance.
Don't spend time on classes; instead, figure out the skills you want to acquire, and learn them yourself. One important reason to sidestep classes is that ... | 23 | https://mathoverflow.net/users/1227 | 36327 | 23,381 |
https://mathoverflow.net/questions/36321 | 8 | Let $X$ be a (irreducible) variety (over $\mathbb{C}$ if necessary, smooth orbifold if necessary), and $U\subset X$ a nonempty open subset, and let $G$ be a finite group with an algebraic action on $U$.
When does the action extend to an action of $G$ on $X$? Are there any conditions that are reasonable to verify, as ... | https://mathoverflow.net/users/622 | Extending group actions on varieties | Well, not always, e.g. take $X=\mathbb P^2$ with homogeneous co-ordinates $x,y,z$, $U$ the locus defined by $xyz\ne 0$ and $\sigma$ the involution of $U$ defined by $(x,y,z)\mapsto (yz,xz,xy)$ (the standard quadratic Cremona transformation of $X$). Then $\sigma$ is of order $2$ and does not extend to $X$. On the other ... | 6 | https://mathoverflow.net/users/8698 | 36337 | 23,388 |
https://mathoverflow.net/questions/36306 | 20 | I am confused about the "bigger picture" when one goes from classical modular forms on $SL\_2(\mathbb{Z})$ and its subgroups to automorphic forms (possibly non-holomorphic).
For classical modular forms we have cusp forms and Eisenstein series. There is the Peterson product defined when one of the forms is a cusp for... | https://mathoverflow.net/users/8688 | Cusp forms and L^2 | This is a long answer because the question asks quite a lot of things. I agree that Gelbart's book, although inspirational, is hard for someone without a strong analytic background. The Boulder and Corvallis proceedings are full of articles which are worth studying if you want to get an understanding of automorphic the... | 29 | https://mathoverflow.net/users/5480 | 36343 | 23,392 |
https://mathoverflow.net/questions/36345 | 34 | I'm wrapping up a summer project that involved a computation in Morava $E$-theory. As background knowledge I had to look into how the Johnson-Wilson theories $E(n)$ and Morava $K$-theories were constructed. This was manageable since I'd been part-way down that road already and there's lots of support in, for example, t... | https://mathoverflow.net/users/1094 | Construction of Morava E-theory | So far as what Morava E-theory should be: Morava E-theory *always* implicitly comes with a choice of a perfect residue field of positive characteristic and a formal group law of finite height over this field. Sometimes people take a very specific formal group law, but there is no reason to be restrictive. The underlyin... | 31 | https://mathoverflow.net/users/360 | 36349 | 23,394 |
https://mathoverflow.net/questions/36299 | 11 | Was providing an alternative proof of the **PNT** one of the main impulses that led to the *discovery* of the Tauberian theorem of Wiener and Ikehara or the other way around?
In any case, do you know who was the **first** individual to realize that, in order to prove the **PNT**, all one needs to have is the non-vani... | https://mathoverflow.net/users/1593 | The Wiener-Ikehara approach to the PNT | This is an elaboration on my comment below John's answer. Its goal is to give a very brief summary of the history underlying the question; I hope that it is more or less correct.
I think that it is fair to say that from the beginning it was understood that non-vanishing on the line $\Re(s) = 1$ was the main requirem... | 5 | https://mathoverflow.net/users/2874 | 36354 | 23,396 |
https://mathoverflow.net/questions/36314 | 2 | I've been hearing about using the sign of the real part of eigenvalues of the multivariate derivative of the change vector field of an ODE at an equilibrium point to determine whether that equilibrium is stable or unstable. Unfortunately, the source is not very good at things like getting the statement of theorems corr... | https://mathoverflow.net/users/nan | Linearization at equilibrium points | I got curious about "the non-diagonalizable case" with purely imaginary eigenvalues
and came up with the system
$$ y' = A y $$ where
$$
A \; = \; \left( \begin{array}{rrrr}
0 & -1 & 1 & 0 \\\
1 & 0 & 0 & 1 \\\
0 & 0 & 0 & -1 \\\
0 & 0 & 1 & 0
\end{array}
\right) .
$$
Taking the matrix of eigenvectors and "gene... | 1 | https://mathoverflow.net/users/3324 | 36355 | 23,397 |
https://mathoverflow.net/questions/36358 | 13 | I wonder if anybody can help me with this problem.
I'm trying to compute the Mertens function for large $n$. The most obvious algorithm is just to compute all primes up to $\sqrt{n}$ and then to sieve. That takes at least an order of $n\log n$ operations, and really even more.
The most recent article that I could ... | https://mathoverflow.net/users/8701 | Computing the Mertens function | [This](https://projecteuclid.org/journals/experimental-mathematics/volume-5/issue-4/Computing-the-summation-of-the-M%C3%B6bius-function/em/1047565447.full) article presents an algorithm to compute Mertens function in $O(x^{2/3}(\log \log x)^{1/3})$ time and $O(x^{1/3}(\log \log x)^{2/3})$ space, I wonder if it is the s... | 11 | https://mathoverflow.net/users/2384 | 36360 | 23,399 |
https://mathoverflow.net/questions/36350 | 5 | My attempts to search via Google seem to be failing, so I thought of asking here.
All the derivatives of the function
$r\_n(z):=\frac{J\_n(z)}{J\_{n-1}(z)}$
where $J\_n(z)$ is the Bessel function of the first kind are expressible in terms of $r\_n(z)$, for instance $\frac{\mathrm{d}}{\mathrm{d}z}r\_n(z)=r\_n(z)^2... | https://mathoverflow.net/users/7934 | Differential equation for a ratio of consecutive Bessel functions | No, you will not find a linear ordinary differential equation (with polynomial coefficients) for $r\_n$. This is because $J\_{n-1}$ has infinitely many zeroes which are not cancelled out by the zeroes of $J\_n$, so that $r\_n$ has infinitely many poles. Holonomic functions, aka solutions of LODEs with polynomial coeffi... | 8 | https://mathoverflow.net/users/3993 | 36374 | 23,407 |
https://mathoverflow.net/questions/36127 | 13 | Let $(A,\mathfrak{m})$ be a local Artinian ring with
finite residue field, which I'm happy to assume is $\mathbf{F}\_3$.
(In particular, $A$ has finitely many elements.)
I would like to do some computations of the following kind, as $I$ ranges over
all of the ideals of $A$.
(0) A way to enumerate all the ideals of ... | https://mathoverflow.net/users/nan | Computational Question about finite local rings: | This is fleshed out from comments of Sam Lichtenstein; Some (or all) of what I have written is surely not the most elegant programming (at best), feel free to improve. One can do the following with MACAULAY2 (if you copy and paste this into the MACAULAY2 prompt it should work:)
`R = GF(2)[x,y]/(x^3,y^3);
m = idea... | 5 | https://mathoverflow.net/users/nan | 36375 | 23,408 |
https://mathoverflow.net/questions/36362 | 20 | Let me admit right at the outset that I have a very superficial outsider's knowledge of homotopy theory. Nevertheless, I was trying to gain some understanding of Hopkins' ICM lecture ['algebraic topology and modular forms.'](http://arxiv.org/abs/math/0212397)
In section 6, he mentions two constructions. To a map
... | https://mathoverflow.net/users/1826 | Characteristic power series for maps of E_{\infty} ring spectra | In short, the series $K\_\phi$ is the "Hirzebruch characteristic series" which arises in the construction/calculation of genera, and in Hirzebruch-Riemann-Roch. The first few chapters of *Manifolds and modular forms* by Hirzebruch et al. describe the classical version of this pretty well.
If I have a one dimensional ... | 16 | https://mathoverflow.net/users/437 | 36381 | 23,412 |
https://mathoverflow.net/questions/36379 | 31 | Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the same convergent sequences?
| https://mathoverflow.net/users/7005 | Is a topology determined by its convergent sequences? | In a metric (or metrizable) space, the topology is entirely determined by convergence of sequences. This does not hold in an arbitrary topological space, and Mariano has given the canonical counterexample. This is the beginning of more penetrating theories of convergence given by [nets](http://en.wikipedia.org/wiki/Net... | 33 | https://mathoverflow.net/users/1149 | 36382 | 23,413 |
https://mathoverflow.net/questions/36319 | 2 | Often in theorems of pcf theory one has the assumption that the length of sequences of functions has to respect some bound so things can be well-defined. For instance, let $a=[\aleph\_2,...,\aleph\_n,...:n<\omega]$ be a set of regular cardinals, say you have a sequence $f\_\beta$ in $\prod a$ of length at most $|a|^+$.... | https://mathoverflow.net/users/3859 | An assumption in pcf theory | Let $a$ be a set of regular cardinals. An element of $\prod a$ is a function $f:a \to \sup a$ such that $f(\kappa) < \kappa$ for every $\kappa \in a$. Suppose we are given a family $f\_i$, $i \in I$, of elements of $\prod a$. In order to ensure that $\sup\_{i \in I} f\_i \in \prod a$ we need to make sure that $\sup\_{i... | 3 | https://mathoverflow.net/users/2000 | 36383 | 23,414 |
https://mathoverflow.net/questions/36296 | 6 | Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have almost identical curvature profiles. I would like to prove a concrete estimate on the total difference of their curvatures in ... | https://mathoverflow.net/users/238 | How does curvature change under perturbations of a Riemannian metric? | This is a straightforward consequence of the fact that $K(x)$ is a continuous function of $g(x)$, $\partial g(x)$, and $\partial^2(g)$.
| 5 | https://mathoverflow.net/users/613 | 36385 | 23,416 |
https://mathoverflow.net/questions/6444 | 14 | Let $R\_n$ be a simple random walk with $R\_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R\_T|$ for some positive $k$.
What is an expression for the probability distribution of $T$?
| https://mathoverflow.net/users/2003 | How long for a simple random walk to exceed $\sqrt{T}$? | For a Brownian motion, [Novikov](http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=TPRBAU000016000003000449000001&idtype=cvips&gifs=Yes) finds an explicit expression for any real moments (positive and negative) of the random variable $(\tau(a,b,c)+c)$, where
$$
\tau(a,b,c) = \inf(t \geq 0, W(t) \leq -a ... | 7 | https://mathoverflow.net/users/3401 | 36413 | 23,432 |
https://mathoverflow.net/questions/36405 | 19 | The Prime Number Theorem was originally proved using methods in complex analysis. Erdos and Selberg gave an elementary proof of the Prime Number Theorem. Here, "elementary" means no use of complex function theory.
Is it possible that any theorem in number theory can be proved without use of the complex numbers?
On ... | https://mathoverflow.net/users/8358 | Complex and Elementary Proofs in Number Theory | Yes, there is a theorem to this effect by Takeuti given in his book "Two applications of logic to mathematics". He shows roughly that complex analysis can be developed in a conservative extension of Peano arithmetic.
| 17 | https://mathoverflow.net/users/51 | 36414 | 23,433 |
https://mathoverflow.net/questions/35560 | 4 | This question is related to the previous discussion [here](https://mathoverflow.net/questions/16393/finding-a-cycle-of-fixed-length).
Due to the [result](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.3377) of Noga Alon et al., there is an $O((2k)^kn)$ algorithm for deciding whether a planar graph $G$ con... | https://mathoverflow.net/users/4248 | Finding a subgraph with slightly large size in planar graphs | I remember thinking about this a while ago, and stopped because it seemed unlikely that $log^2 n$ paths can be found in polynomial time. This was my argument, if I remember correctly.
The best known algorithm for Hamiltonian path is $O^\*(2^n)$. I think improving this to sub-exponential time, like $O(2^{o(n)})$ woul... | 4 | https://mathoverflow.net/users/8075 | 36415 | 23,434 |
https://mathoverflow.net/questions/36403 | 12 | Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ is the unit ball w.r.t the norm?
This seems to me like a fundamental question but I cannot seem to find anything. Any ... | https://mathoverflow.net/users/4047 | What is known about the Gaussian measure of the unit ball in a Hilbert Space? | You can't talk about "the" Gaussian measure on an infinite-dimensional Hilbert space, for the same reason that you can't talk about a uniform probability distribution over all integers. It doesn't exist; see Richard's answer. However, there are a lot of non-uniform Gaussian measures on infinite dimensional Hilbert spac... | 10 | https://mathoverflow.net/users/2294 | 36417 | 23,436 |
https://mathoverflow.net/questions/36396 | 13 | Suppose that $(A,\mathfrak{m})$ is a local Artinian ring.
If $A$ is Gorenstein, then $A$ admits a dualizing functor
on finite length modules defined by $D(M):= Hom\_A(M,A)$ which preserves
lengths. If $M$ is a finite length $A$ module, let $\mathrm{length}(M)$
denote the length of $M$.
If $I$ is an ideal of $A$ and $J ... | https://mathoverflow.net/users/nan | Length of I/I^2 versus Ann(I)/Ann(I)^2 in Artinian rings. | UPDATE 10/04/10:
Here are some partial results in the graded case. Let $R=\oplus\_0^s R\_i$ be a graded Gorenstein algebra over $k=R\_0$ ($s$ is the socle degree). $R$ is said to have *strong Lefschetz property* (SLP) if for a general linear form $l$ in $R$, the multiplication map $\times l^a: R\_i \to R\_{i+a}$ has ... | 9 | https://mathoverflow.net/users/2083 | 36419 | 23,437 |
https://mathoverflow.net/questions/36420 | 12 | Let a problem instance be given as $(\phi(x\_1,x\_2,\dots, x\_J),M)$ where $\phi$ is a diophantine equation, $J\leq 9$, and $M$ is a natural number. The decision problem is whether or not a given instance has a solution in natural numbers such that $\sum\_{j=1}^J x\_j \leq M$. With no upper bound M, the problem is unde... | https://mathoverflow.net/users/8719 | Is the solution bounded Diophantine problem NP-complete? | A particular quadratic Diophantine equation is NP-complete.
$R(a,b,c) \Leftrightarrow \exists X \exists Y :aX^2 + bY - c = 0$
is NP-complete. ($a$, $b$, and $c$ are given in their binary representations. $a$, $b$, $c$, $X$, and $Y$ are positive integers).
Note that there are trivial bounds on the sizes of $X$ and... | 20 | https://mathoverflow.net/users/8723 | 36424 | 23,441 |
https://mathoverflow.net/questions/36410 | 0 | How to prove $U \otimes Ind W = Ind(Res(U) \otimes W)$? where U is a representation of G and W is a rep of H, a subgroup of G. $Ind(W)$ is the induced rep and $Res(U)$ is the restrict rep.
I got the answer, by both approaches: groups algebra and constructing isomorphic map. Thanks for all the very helpful comments an... | https://mathoverflow.net/users/5737 | How to prove $U \otimes Ind W = Ind(Res(U) \otimes W)$ | This is really a comment which got too long.
Personally, I always find this one rather confusing. If you think in terms of modules over group rings, we want to show that $U \otimes (\mathbb{C}[G]\otimes\_{\mathbb{C}[H]} W) \cong \mathbb{C}[G] \otimes\_{\mathbb{C}[H]} (U\otimes W)$. The $G$-equivariant isomorphism is ... | 4 | https://mathoverflow.net/users/4042 | 36428 | 23,445 |
https://mathoverflow.net/questions/36392 | 5 | This is a direct consequence of my previous question: [Extending group actions on varieties](https://mathoverflow.net/questions/36321/extending-group-actions-on-varieties)
In his answer, inkspot said that group actions can be extended if the variety has ample canonical class and is smooth, but mentions that canonical... | https://mathoverflow.net/users/622 | Are orbifold singularities canonical? | For quotient singularities there is the so-called Reid-Tai criterion to check whether the singularity is canonical or not.
Suppose $G$ is a finite subgroup of $GL\_n(\mathbb{C})$ without quasi-reflections. Let $m=|G|$ and fix a primitive $m$-th root of unity $\zeta$. Let $g\in G$ and let $0\leq a\_i < m$ be such that $... | 11 | https://mathoverflow.net/users/8621 | 36430 | 23,447 |
https://mathoverflow.net/questions/35567 | 16 | If I have committed to a number x by revealing g^x mod p, can I prove that 0 < x mod (p-1) < (p-1)/2, i.e. that x is positive, without leaking any more information about x?
My bounty is ending in 4 days and I am unsatisfied with the current answers so I would like to provide more context and also expand the question ... | https://mathoverflow.net/users/2003 | Zero-knowledge proof of positivity | This answer combines my three comments to the question and expands them a little.
Following [BM84], let’s call the integers $g^x \bmod p$ for $0 < (x \bmod (p−1)) < (p−1)/2$ *principal square roots*. We call the problem of deciding, given $p$, $g$ and $y$, whether an integer $y$ is a principal square root or not the ... | 11 | https://mathoverflow.net/users/7982 | 36433 | 23,449 |
https://mathoverflow.net/questions/36418 | 2 | My specific situation is that I have a non-spacelike continuous future directed curve $\gamma:[0,a)\to M$ in a Lorentzian manifold. The curve must necessarily satisfy a local Lipschitz condition and therefore the derivative of the curve exists almost everywhere. In these circumstances, as I understand it, one can show ... | https://mathoverflow.net/users/5993 | Reference for existence and uniqueness of differential equations for low differentiability? | I do not know any recent book on ODE's satisfying yours criterium. The classical books below do a fairly good job though. The first being more comprehensive and the second more elementary(no measure theory needed).
1. Coddington, Earl A.; Levinson, Norman.
Theory of ordinary differential equations.
2. Petrovski, I. ... | 2 | https://mathoverflow.net/users/605 | 36436 | 23,451 |
https://mathoverflow.net/questions/36388 | 8 | Complex moduli space (or Teichmuller space) of a Quintic Calabi-Yau 3-fold is a
101-dimensional complex orbifold. Does it have a toric structure?
| https://mathoverflow.net/users/5259 | Is the complex moduli of Quintic Calabi-Yau toric? | The complex moduli space does not admit a toric strucutre, since the orbifold fundamental group of a toric orbifold must be abelian. Indeed, $\pi\_1(\mathbb C^\*)^n$ surjects on the orbifold fundamental group. Also, the orbifold stabisier of each point on a toric orbifold
is a finite abelian group. At the same time th... | 7 | https://mathoverflow.net/users/943 | 36438 | 23,452 |
https://mathoverflow.net/questions/36445 | 8 | Given a coherent $\mathcal{D}\_X$-Module $M$, one can assign to it its characteristic variety
$$ch(M)\subseteq T^\*X$$
or one could look at its support
$$supp(M)\subseteq X$$
as an $\mathcal {O}\_X$ module. Is there a relation between these two spaces?
Edit: Is it true, that the characteristic variety of a holonomi... | https://mathoverflow.net/users/2837 | Relation between characteristic variety and support of D-Module | I'm not an expert either but, at least for holonomic D-modules, the relation you're looking for should be
$$
supp(M) = ch(M) \cap T^\*\_X X
$$
where $T^\*\_XX$ is the zero section of the cotangent bundle identified with $X$.
You can check it in the basic examples: $X = \mathbb{C}$, $M$ the trivial bundle or the $... | 8 | https://mathoverflow.net/users/1985 | 36456 | 23,460 |
https://mathoverflow.net/questions/36457 | 4 | My question: is the set of potential games closed under convex combinations?
An n player game with action set $A = A\_1 \times \ldots \times A\_n$ and payoff functions $u\_i$ is called an exact potential game if there exists a potential function $\Phi$ such that:
$$\forall\_{a\in A} \forall\_{a\_{i},b\_{i}\in A\_{i}}... | https://mathoverflow.net/users/3027 | Is the convex combination of two potential games a potential game? | No, the set of ordinal potential games is not convex. I will think about an example, but it follows from a little fact that I've yet to get around to publishing:
Theorem: Any convex set of games strictly containing the exact potential games contains a game with no pure Nash equilibrium.
In particular, since every o... | 6 | https://mathoverflow.net/users/5963 | 36465 | 23,467 |
https://mathoverflow.net/questions/36443 | 3 | Let $K$ be a field (of course, of positive characteristic, unless you want a trivial question). Let $G$ be a finite group, and $V$ and $W$ be two completely reducible (finite-dimensional) representations of $G$ over $K$. Is the (interior) tensor product $V\otimes\_K W$ a completely reducible representation of $G$ ?
I... | https://mathoverflow.net/users/2530 | Innocent question on tensor products of modular representations | Jim has already given the correct answer "no".
Here is a hopefully instructive example. Let $k$ be an alg. closed field of pos. char $p$, and let $G = SL\_2(k)$. Write $V = k^2$ for the "natural" 2-dimensional representation of $G$ say with basis $e\_1,e\_2$. Let $W = S^pV$ be the $p$-th symmetric power of $V$. Then... | 7 | https://mathoverflow.net/users/4653 | 36473 | 23,471 |
https://mathoverflow.net/questions/36444 | 42 | Can anyone give me a plain-and-simple
definition of an E-infinity algebra without using
the words "operad," "ring spectrum," or
"stable homotopy"?
Sorry, but I honestly couldn't find it using
all on-line resources at my disposal.
Thanks!
| https://mathoverflow.net/users/1186 | Definition of an E-infinity algebra | In characteristic 0, Kadeishvili has a notion of $C\_{\infty}$ algebra which models rational homotopy theory. See the last paragraph of the introduction of his paper arXiv:0811.1655. His point of view is to simply consider $A\_{\infty}$ algebras whose operations satisfy a certain property with respect to shuffle maps. ... | 14 | https://mathoverflow.net/users/6948 | 36478 | 23,475 |
https://mathoverflow.net/questions/36479 | 6 | Let f:X→Y be proper birational morphism between two quasi-projective varieties over an algebraically closed field $k$. I am particularly interested in the case where the characteristic of $k$ is positive; Y is singular, X is smooth and both X and Y are not projective.
Let D be a closed subset of Y, let E=f-1(D) and a... | https://mathoverflow.net/users/8621 | Exact sequence in étale cohomology related with proper birational morphism | Let $V := X\setminus E$ and $U := Y\setminus D$ and $j\colon U \rightarrow Y$ and
$k\colon V \rightarrow X$ the inclusions. We have exact sequences
$\cdots\rightarrow H^\ast(X,k\_!\mathbb Z\_\ell)\rightarrow H^\ast(E,\mathbb
Z\_\ell)\rightarrow H^\ast(X,\mathbb Z\_\ell)\rightarrow\cdots$ and a similar one for
$Y$, $D$ ... | 8 | https://mathoverflow.net/users/4008 | 36482 | 23,476 |
https://mathoverflow.net/questions/36481 | 1 | I've got what might be a couple of very basic questions on the fundamental representations of locally isomorphic semi-simple Lie groups and their relationship to representations of the corresponding Lie algebra. I know (from reading Cornwell, 'Group Theory in Physics') that every representation of a semi-simple Lie alg... | https://mathoverflow.net/users/7317 | Reps of groups and reps of algebras | Let us consider for simplicity the complex semi-simple case. As you mention, every irreducible representation $\rho$ of a semi-simple finite dimensional Lie algebra $g$ integrates to a representation of the corresponding simply connected Lie group $G$. But even if we suppose $\ker\rho=0$, the representation of $G$ may ... | 2 | https://mathoverflow.net/users/2349 | 36490 | 23,481 |
https://mathoverflow.net/questions/36502 | 3 | In a round-robin tournament with $n$ teams, each team plays every other team exactly once. Thus, there are $n(n-1)/2$ total games played. How many different standings can result? By a "standing" I mean the ordered sequence $(W\_1, \ldots, W\_n)$ where $W\_i$ is the number of wins by the $i$th player. Assume that no gam... | https://mathoverflow.net/users/4758 | Round-Robin Tournaments and Forests | The bijection between score vectors and forests on labeled nodes is due to Kleitman and Winston. ([This](https://doi.org/10.1007/BF02579176 "Kleitman, D.J., Winston, K.J. Forests and score vectors. Combinatorica 1, 49–54 (1981). zbMATH review at https://zbmath.org/?q=an:0491.05028") paper)
A small clarification, your... | 9 | https://mathoverflow.net/users/2384 | 36503 | 23,488 |
https://mathoverflow.net/questions/36348 | 7 | Let $(a\_{mn})\_{m,n\in\mathbb{N}}$ and $(b\_m)$ be sequences of complex numbers.We say that $(a\_{mn})$ and $(b\_m)$ constitute *an infinite system of linear equations in infinitely many variables* if we seek a sequence $(x\_n)$ of complex numbers such that $\forall m\in\mathbb{N}:$ $\sum\_{n=1}^{\infty}a\_{mn}x\_n=b\... | https://mathoverflow.net/users/1849 | infinitely many linear equations in infinitely many variables | The systems of this kind are fairly common in applications. For example, they naturally appear when solving boundary value problems for linear partial differential equations using the method of separation of variables.
Predictably, the problem is not meaningful for *any* sequences {$a\_{nm}$}, {$b\_m$}, but only for ... | 10 | https://mathoverflow.net/users/8670 | 36504 | 23,489 |
https://mathoverflow.net/questions/18938 | 7 | Every partially ordered set gives a triangulation of (the geometric realisation of) its order complex. (The n-simplices of the order complex are the chains $x\_0\leq x\_1\leq\cdots\leq x\_n$.) However, there are triangulations of topological spaces that do not arise this way.
Is there a name for triangulations having... | https://mathoverflow.net/users/1291 | Triangulations coming from a poset. Or: What conditions are necessary and sufficient for a finite simplicial complex to be the order complex of a poset? | Here are necessary and sufficient conditions for an abstract, finite simplicial complex $\mathcal{S}$ to be the order complex of some partially ordered set.
(i) $\mathcal{S}$ has no missing faces of cardinality $\geq 3$; and
(ii) The graph given by the edges (=$1$-dimensional simplices) of $\mathcal{S}$ is a compar... | 13 | https://mathoverflow.net/users/8735 | 36505 | 23,490 |
https://mathoverflow.net/questions/36466 | 16 | Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by:
$F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative structure.
Can we define $H^i(X,F)$ ? Note that for $N=1$, this would be just $H^i(X,O\_X^\*)$.
(Please give reference for ... | https://mathoverflow.net/users/5259 | Do we have non-abelian sheaf cohomology? | The quick reply is: not really for $i \gt 2$, and not in the way you perhaps expect for $i=2$, see below.
---
**EDIT (Feb 2017):** Debremaeker's PhD thesis [0] has now been translated into English and placed on the arXiv: [Cohomology with values in a sheaf of crossed groups over a site](https://arxiv.org/abs/1702... | 42 | https://mathoverflow.net/users/4177 | 36508 | 23,493 |
https://mathoverflow.net/questions/36528 | 3 | In the plane, the exterior angle of a vertex is $\pi -$ the standard ("interior") angle, which may be negative in some cases. The following is true for non-weird polygons:
>
> The sum of the exterior angles at each vertex is a full turn ($2\pi$ radians).
>
>
>
I am informally calling polygons with self-crossin... | https://mathoverflow.net/users/2498 | An exterior angle theorem for n-dimensional polytopes? | What you call exterior angles are the curvatures at the vertices, and the result you are referring to is the combinatorial Gauss-Bonnet theorem. It says that in an angled 2-complex $K$ the sum of the curvatures at the vertices plus the sum of the curvatures at the polygonal faces is $2\pi \chi(K)$ (Euler characteristic... | 1 | https://mathoverflow.net/users/2384 | 36534 | 23,507 |
https://mathoverflow.net/questions/36511 | 4 | Let $M$is a closed oriented 2n-dimensional smooth manifold, $E$ is a 2n-dimensional oriented real vector bundle on $M$, with inner product on each fibers.
Let $\tau=(\sqrt{-1})^{n}c(e\_{1})c(e\_{2})...c(e\_{2n})$, use it we can define a $\mathbb{Z}\_2$-grading on $\wedge(E^{\*})\otimes\mathbb{C}=\wedge\_{+}(E^{\*})\oti... | https://mathoverflow.net/users/3896 | A question about a formula of Pfaffian | The supertrace can be evaluated either by Berezin Gaussian integration, or equivalently by
summation over a Clifford algebra. Here is a description of the second method.
Let $\omega$ be a skew-symmetric 2n by 2n matrix. Let $\{{\ e\_1, e\_2, . . . . e\_{2n} \}}$ be a real
2n-dimensional Clifford algebra. Then:
$ex... | 2 | https://mathoverflow.net/users/1059 | 36551 | 23,515 |
https://mathoverflow.net/questions/36552 | 0 | Disclaimer: This is not a homework problem. I stumbled on this puzzle on internet and I also have the answer. However I am not able to figure out whats the method to be used to arrive at the answer.
The puzzle is as below:
The product of the ages of David's children is the square of the sum of their ages. David has... | https://mathoverflow.net/users/8246 | How to tackle this puzzle? | Well, we know that the sum is at most $14+13+12+11+10+9+8+7=84$, so the product is at most $7056$.
If there are $7$ or more children, then the product is at least $8!>7056$, so there are at most $6$ children.
Furthermore, if there are $6$ children, the sum is at most $84-8-7=69$, so the product is at most $69^2=476... | 2 | https://mathoverflow.net/users/1355 | 36558 | 23,518 |
https://mathoverflow.net/questions/36550 | 1 | Hey I'm trying to understand what kind of boundary data I can pose for the wave equation. Let's work in one dimension for now. It appears I should be able to pose any Neumann, Dirichlet or Robin boundary conditions.
I've heard that you need your boundary data to be 'consistent with the Cauchy data'.
I would like to ... | https://mathoverflow.net/users/8755 | Can I pose any bounary data for the wave equation on $[0,\infty)$ for given Cauchy data? | I am just going to answer here the problem actually posed in the title.
The one dimensional wave equation can be re-written as $\partial\_u\partial\_v \phi = 0$, where $u$ and $v$ are the null variables $x + t$ and $x - t$. The initial data then is prescribed on $u+v \geq 0, u-v = 0$, while the boundary is $u+v = 0,... | 2 | https://mathoverflow.net/users/3948 | 36561 | 23,520 |
https://mathoverflow.net/questions/36269 | 1 | Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^\*$ be the dual sheaf.
How can I see that the determinant map is a map from $A^\*$ to $O\_X((1-\frac{1}{e})C)$?
And how to unde... | https://mathoverflow.net/users/3233 | Q-Divisor and Determinant Map on a Maximal Order | The determinant should map to ${\cal A}^\* \rightarrow {\cal O}\_X(e(1-1/e)$. You can see this along $C$ in codimension one since if you \'etale localize at the generic point of $C$ then the structure Theorem for maximal orders says that ${\cal A}$ localizes to something Morita
equivalent to
R tR ... tR
R R ... tR... | 1 | https://mathoverflow.net/users/8762 | 36567 | 23,523 |
https://mathoverflow.net/questions/36574 | 0 | Given an unweighted graph $G = (V, E)$, let the cut function on this graph be defined to be:
$C:2^V \rightarrow \mathbb{Z}$ such that:
$$C(S) = |\{(u,v) \in E : u \in S \wedge v \not\in S\}|$$
Suppose you have the ability to query $C$, but otherwise have no knowledge of the edge set $E$. Is it possible to reconstruct ... | https://mathoverflow.net/users/3027 | Reconstructing a graph given access to its cut function | Asking individual vertices, you figure out the valence of each vertex with $n$ questions. Asking for pairs of vertices, you can then decide if each pair has an edge between them or not: they have an edge if and only if the "degree" on the pair is two less than the sum of the "degrees". Thus you can reconstruct the grap... | 5 | https://mathoverflow.net/users/8761 | 36578 | 23,530 |
https://mathoverflow.net/questions/36549 | 2 | Usually one shows the density of the functions $\sin(kx)$ in $L^2([0,1])$ using the Fourier transform. This in fact comes from the Stone-Weierstrass theorem however and then uses the density of continuous functions in $L^2([0,1])$.
However, the Stone–Weirstrass theorem can be used to show, for example, that the funct... | https://mathoverflow.net/users/8755 | Coefficients from Stone–Weierstrass versus Fourier transform | Just a comment if you choose coefficients $c\_{k,n}$ such that
$$
\lim\_{n\to\infty} \left(\sum\_{k=-n}^{n} c\_{k,n} e^{2\pi i n x}\right) \to f (x)
$$
in some sense, e.g. $L^1$, then these are not unique. It is even known that the obvious choice $c\_{k,n} = \hat{f}(k) = \int e^{-2\pi i n x} f(x) dx$ is not the best. ... | 7 | https://mathoverflow.net/users/3983 | 36585 | 23,535 |
https://mathoverflow.net/questions/36583 | 3 | Given an unweighted graph $G = (V, E)$, let the cut function on this graph be defined to be:
$C:2^V \rightarrow \mathbb{Z}$ such that:
$$C\_G(S) = |\{(u,v) \in E : u \in S \wedge v \not\in S\}|$$
For any two vertices $i,j \in V$, let the $(i,j)$ min-cut in a graph $G$ be:
$$\alpha\_{i,j}(G) = \min\_{S \subset V : i... | https://mathoverflow.net/users/3027 | To what degree do min-cuts specify the cut function of a graph? | If G and H are graphs with n vertices, Δ(G,H) can be Θ(n2).
Here is an example which shows this. Let n=4k+1 be a prime. Define two graphs G=(V,E) and H=(V,E′) by V={0,1,2,…,4k}, E = {{i,j} | 1 ≤ ((j−i) mod n) ≤ k}, E′ = {{i,j} | k+1 ≤ ((j−i) mod n) ≤ 2k}. Note that G and H are both unions of k edge-disjoint Hamiltoni... | 6 | https://mathoverflow.net/users/7982 | 36589 | 23,537 |
https://mathoverflow.net/questions/36546 | 3 | I am interested in the properties of the following subclass of split graphs:
The class consists of all split graphs $G=(C\cup I)$ where $C$ is a clique and $I$ an independent set, and *every* pair of vertices in $I$ have at least one common neighbor in $C$.
Does this class of graphs have a special name? Has this c... | https://mathoverflow.net/users/1667 | Have this subclass of split graphs been studied before? | Split graphs of diameter 2?
| 7 | https://mathoverflow.net/users/7170 | 36591 | 23,539 |
https://mathoverflow.net/questions/36588 | 0 | While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order mixed partial derivative term is denoted by
$
2\frac{{\partial ^4 u}}{{\partial ^2 x\partial ^2 y}} = \Delta \_{xy}^4 u^{k... | https://mathoverflow.net/users/7875 | Delta notation used for describing numerical stencil | If I saw that symbol alone, I would guess that $\Delta^2\_x u\_{i,j}:=u\_{i+1,j}-2u\_{i,j}+u\_{i-1,j}$. But with that definition the stencil $\Delta^4\_{xy}u\_{i+1,j+1}$ looks very odd.
| 1 | https://mathoverflow.net/users/1898 | 36598 | 23,544 |
https://mathoverflow.net/questions/36592 | 5 | To be slightly more precise: let $M\subset B(H)$ be a finite von Neumann algebra equipped with a faithful normal trace $\tau$, and let $L^0(M,\tau)$ be the completion of $M$ in the measure topology; this is an algebra, whose elements can be identified with those densely-defined and closed operators on $H$ that are affi... | https://mathoverflow.net/users/763 | range projection of an unbounded idempotent affiliated to a finite von Neumann algebra | I couldn't find it in Kadison & Ringrose. But what about this: let $\xi\in R$ and let $u$ be a unitary in $M'$. Since $e$ is affiliated with $M$, $ue=eu$. So $u\xi=ue\xi=eu\xi\in R$. This shows that $uR\subset R$ for any unitary $u$, and so $uR=R$ for any unitary $u$ in $M'$. This in turn is equivalent to $ur=ru$, wher... | 5 | https://mathoverflow.net/users/3698 | 36605 | 23,548 |
https://mathoverflow.net/questions/36532 | 12 | Suppose you have a labeled tree $T$ on vertices $V=\lbrace 1,\ldots,n\rbrace$ that is drawn uniformly at random from the set of all $n^{n-2}$ such trees. I am seeking an $f$ satisfying the following desiderata:
D1. $f(T)$ is a (random) tree on the vertex set $V'=\lbrace 1,\ldots,n+1\rbrace$;
D2. the distribution of... | https://mathoverflow.net/users/7967 | Is there a simple inductive procedure for generating labeled trees uniformly at random, without direct recourse to Prüfer sequences? | This is an interesting question. For any fixed positive integer $d \geq 2$, write $T\_d^{\infty}$
for the complete infinite rooted $d$-ary tree (by this I mean every node has exactly $d$ children). [Luczak and Winkler](http://onlinelibrary.wiley.com/doi/10.1002/rsa.20011/abstract) proved the existence of a procedure w... | 2 | https://mathoverflow.net/users/3401 | 36606 | 23,549 |
https://mathoverflow.net/questions/35226 | 8 | Let $f:X\to Y$ be a birational morphism, $X, Y$ projective, $X$ smooth (threefold if this helps). Let $Exc(f)\subseteq X$ be the exceptional locus of $f$ and let $E\subseteq Exc(f)$ be an irreducible divisor. Is it true that for any curve $C\subseteq E$ contracted by $f$ one has $C\cdot E<0$? I can see this is true if ... | https://mathoverflow.net/users/5419 | Negativity of contraction | Dear Carlos, the statement is false in general. For example let $Y$ be $\mathbb{C}^3$, let $f\_1 : X\_1 \rightarrow Y$ be the blowup of a point on $Y$, and $f\_2 : X \rightarrow X\_1$ the blowup of a point on the exceptional divisor of $f\_1$. Let $f : X \rightarrow Y$ be the composition. The exceptional locus of $f$ h... | 5 | https://mathoverflow.net/users/8770 | 36612 | 23,553 |
https://mathoverflow.net/questions/36613 | 6 | Suppose that $K$ is a number field, and (writing $G\_K=\mathrm{Gal}(\overline{K}/K)$), suppose that $\phi:G\_K\to \overline{\mathbb{Q}}$ is a finite order character of $G\_K$. I believe that the obstruction to taking a square root of $\phi$ (that is, the obstruction to finding some finite order $\chi:G\_K\to \overline{... | https://mathoverflow.net/users/3513 | obstruction to taking the square root of a Galois character | With luck, this will be blunder-free (and if it's not, please tell me!):
Consider the short exact sequence
$$1 \to \mu\_2 \to \overline{\mathbb Q}^{\times} \to \overline{\mathbb Q}^{\times}
\to 1,$$
with the third arrow being squaring, and with trivial $G\_k$-action. Passing to cohomology,
the sequence of $H^0$s is ... | 12 | https://mathoverflow.net/users/2874 | 36618 | 23,555 |
https://mathoverflow.net/questions/36607 | 0 | Suppose X is compact and totally disconnected space, and that phi a homeomorphism of X.
We say a subset Z of X is phi-invariant if phi(Z) = Z. A phi-invariant set is minimal if it is closed, phi-invariant, nonempty and the smallest of all such sets. We say (X,phi) is minimal if X itself is a minimal set.
An orbit o... | https://mathoverflow.net/users/8769 | Does essentially minimal imply minimal? | As far as I understand the author tried (with missing closure in conditions (1) and (2), as it was already pointed out) to reproduce the standard definition of an essentially minimal dynamical system as one which has a unique minimal subset, see, for instance, Definition 1.2 from
MR1194074 (94f:46096)
Herman, Richar... | 4 | https://mathoverflow.net/users/8588 | 36620 | 23,557 |
https://mathoverflow.net/questions/36611 | 5 | It is known that a valuation domain is a principal ideal ring if and only if its prime ideals are principal. Is it a principal ideal ring when its unique maximal ideal is principal?
| https://mathoverflow.net/users/5775 | Is a valuation domain PID when its maximal ideal is principal? | I assume that by a valuation domain you mean an integral domain $R$ with fraction field $K$ such that: for all $x \in K^{\times}$, at least one of $x,x^{-1}$ lies in $R$.
In this case, I believe the answer is **no**. Let $R$ be any valuation domain whose value group $K^{\times}/R^{\times}$ is isomorphic, as a totally... | 9 | https://mathoverflow.net/users/1149 | 36636 | 23,570 |
https://mathoverflow.net/questions/36639 | 2 | Suppose I have a singular projective variety defined by some homogeneous equations in complex projective space. Is the resolution of singularities effective? That is, do I actually know which smooth centers to blow up?
| https://mathoverflow.net/users/nan | Is resolution of singularities effective? | Yes, in the sense that resolution of singularities is implemented in the computer algebra package Singular. See the manual of Singular for references. (There might be other/better references.) However, if I remember correctly the centers are not unique.
| 5 | https://mathoverflow.net/users/8621 | 36640 | 23,572 |
https://mathoverflow.net/questions/36568 | 6 | To do Algebraic K-theory, we need a technical condition that a ring $R$ satisfies $R^m=R^n$ if and only if $m=n$. I know some counterexamples for a ring $R$ satisfies $R=R^2$.
Are there any some example that $R\neq R^3$ but $R^2 = R^4$ or something like that?
(c.f. if $R^2=R^4$, then we need that $R^3=R^5=\ldots =... | https://mathoverflow.net/users/7776 | Subtle counterexample to $m\neq n$ but $R^m=R^n$ for some ring $R$ ? | Rings that satisfy the condition
$R^n \cong R^m \iff n=m$ are said to have *invariant basis number* or the *invariant basis property*. P. M. Cohn has constructed examples of rings which fail to have this property, even giving examples of (non commutative) integral domains for which e.g. $R^3\cong R$ but $R^2\neq R$.
... | 4 | https://mathoverflow.net/users/6701 | 36644 | 23,574 |
https://mathoverflow.net/questions/36625 | 2 | Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L\_{poly}\mathbb{C}^\times$ of loops that look like $w\_0 + w\_1z +w\_2z^2 + \cdots + w\_nz^n$ where $z = e^{i\theta}$ (as Andrew notes below this is not a group because its not closed under taking inverses). Eq... | https://mathoverflow.net/users/7 | parameterizing polynomial loops in $\mathbb{C}^\times$ | This is mostly a series of comments, but guided by the questions you asked.
First of all, I will only talk about $X\_n$, interpreting it as the space of non-zero complex polynomials $p$ of degree at most $n$ such that no root of $p$ lies on the unit circle, taken up to non-zero scaling. We may as well think of the po... | 5 | https://mathoverflow.net/users/4344 | 36651 | 23,581 |
https://mathoverflow.net/questions/36658 | 4 | Hi,
let $E$ be a vector bundle over a smooth projective variety $X$ and $\pi:\mathbb{P}(E)\rightarrow X$ its projectivization, $T\_{\pi}:=ker(\pi\_{ \* })$ where $\pi\_{\*}:T\_{\mathbb{P}(E)}\rightarrow \pi^{\*}T\_X$, let $\mathcal{O}\_E(-1)\hookrightarrow\pi^{\*}E$ be the "tautological" bundle over $\mathbb{P}(E)$.... | https://mathoverflow.net/users/4971 | Which is the correct generalization of Euler sequence to the projectivization of a vector bundle? | This sequence is indeed exact. Once you check that the maps are globally defined, exactness can be checked fiberwise (remember that we are dealing with a sequence of *vector bundles*, not an arbitrary sequence of sheaves), and in this case it follows by the usual Euler sequence for the projective space.
| 5 | https://mathoverflow.net/users/828 | 36661 | 23,586 |
https://mathoverflow.net/questions/36494 | 3 | Maybe this is too technical and elementary, but I cannot make up my mind, nor find a reference.
The situation is the following: let $X$ be a double cochain (right half-plane) complex of abelian groups and let
$$
(\mathbf{Tot}^{\prod} X)^n = \prod\_{p+q = n}X^{pq}
$$
denote its total-product complex. The first fi... | https://mathoverflow.net/users/1246 | Is the first filtration Hausdorff? | No, this filtration is not necessarily Hausdorff. The problem is connected to nonexactneess of the inverse limit functor. Here is a family of examples to illustrate the basic issue.
Suppose $\cdots \to A\_2 \to A\_1 \to A\_0$ is an inverse system of abelian groups. Define a complex by
$$
X^{p,-p} = X^{p+1,-p} = A\_p
... | 2 | https://mathoverflow.net/users/360 | 36662 | 23,587 |
https://mathoverflow.net/questions/36642 | 15 | My question is about $m \times n$ binary matrices (aka $\{0,1\}$-matrices), whose rows all sum to the same value, and whose columns all sum to the same value (but these two values may be different).
The first question is simply: is there a standard name for such matrices? They correspond to the biadjacency matrices o... | https://mathoverflow.net/users/7767 | Binary matrices with constant row and column sums | To answer your question about interesting combinatorial objects: Your Sylvester-Hadamard matrix example generalizes in at least two ways.
1. The incidence matrix of any balanced incomplete [block design](https://en.wikipedia.org/wiki/Block_design) or, more generally, $t$-design has constant row and column sums. Speci... | 8 | https://mathoverflow.net/users/484 | 36668 | 23,591 |
https://mathoverflow.net/questions/36665 | 7 | I have a triangle $ABC$ with side lengths $a,b,c$ (edges $BC, CA, AB$ respectively).
I have a point $p$ with barycentric coordinates $u:v:w$.
These are normalised: $u+v+w=1$.
$1:0:0$ corresponds to point $A$, $0:1:0$ is $B$ etc.
Is there a simple expression for the distance $d$ of the point $p$ from $A$ ?
(My i... | https://mathoverflow.net/users/3519 | Distance of a barycentric coordinate from a triangle vertex | There isn't really a simple formula, but you can use vector methods.
Let $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ be position vectors
of the vertices. A point $P$ with normalized barycentric coordinates
$(u,v,w)$ has position vector $\mathbf{p}=u\mathbf{a}+v\mathbf{b}+w\mathbf{c}$.
Therefore $\mathbf{p}-\mathbf{a}=v... | 4 | https://mathoverflow.net/users/4213 | 36669 | 23,592 |
https://mathoverflow.net/questions/36674 | 1 | I'm looking for a proof in the literature of the following fact:
Let $A\_t$ be a $C^1$-function of one argument $t \in (a,b)$ taking values in the self-adjoint $N \times N$ matrices. Suppose that for every $t$ the spectrum of $A\_t$ is simple. Denote by $\lambda\_t$ a continuous parametrization of an eigenvalue of $A\_... | https://mathoverflow.net/users/3983 | Parametrizing eigenvectors | I do not know a reference, but here is an easy argument. Consider the space $\overline{M}$ of pairs $(A,\lambda)$ where $A$ is a (self-adjoint) $N \times N$ matrix and $\lambda$ is a root of the characteristic polynomial of $A$. Over the open subset of $\overline{M}$ lying above the operators with distinct eigenvalues,... | 3 | https://mathoverflow.net/users/4344 | 36677 | 23,597 |
https://mathoverflow.net/questions/36638 | 13 | Let $f(x\_1,\ldots, x\_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?
| https://mathoverflow.net/users/nan | Effective algorithm to test positivity | If by effective you mean "is this computable", then yes, the computational versions of Tarski-Seidenberg such as cylindrical algebraic decomposition give you a finite algorithm. (I suppose this is assuming your polynomial has rational coefficients, or at least algebraic coefficients each given by a polynomial they sati... | 14 | https://mathoverflow.net/users/5963 | 36685 | 23,602 |
https://mathoverflow.net/questions/10029 | 14 | Let $k$ be a number field and $F$ a $1$-variable function field over $k$ (a finitely generated extension of $k$, of transcendence degree $1$, in which $k$ is algebraically closed). If $F$ becomes the rational function field over every completion $k\_v$ of $k$, then $F$ is the rational function field over $k$. This is a... | https://mathoverflow.net/users/2821 | A local-to-global principle for being a rational surface | It seems to me that there are irrational surfaces over $\mathbb Q$ that are $\mathbb Q\_v$-rational for all $v$. (I couldn't find them in the literature, but didn't look very hard. Almost certainly they are to be found there, in papers by either Iskovskikh or Colliot-Thelene.)
Take the affine surface $S$ given by $y... | 5 | https://mathoverflow.net/users/8726 | 36687 | 23,604 |
https://mathoverflow.net/questions/35092 | 2 | A [Median graph](http://en.wikipedia.org/wiki/Median_graph) is graph with the property, that for each three vertices $x,y,z$ there is a unique vertex $m(x,y,z)$ lying on shortest paths from $x$ to $y$, from $y$ to $z$ and from $z$ to $x$. Examples are trees, the Cayley graph of $\mathbb{Z}^n$ (with the standart generat... | https://mathoverflow.net/users/3969 | Stability of medians in Median graphs | Here is an attempt to prove that the answer is **yes**.
**Claim 1**: Median graphs are bipartite.
This surely appears in the literature and is easy to verify. (Consider for a contradiction the shortest odd cycle and a median of 3 vertices on it: a pair of adjacent ones and a third one "opposite" of this pair.)
... | 4 | https://mathoverflow.net/users/8733 | 36702 | 23,611 |
https://mathoverflow.net/questions/36691 | 11 | A configuration of queens on an 8 by 8 chessboard (or n by n if you like) is a *queen domination* if every square on the board lies in the same row, column, or diagonal as at least one of the queens. The Queens Domination Problem is to find the minimum number of queens necessary for a queen domination. A *solution* to ... | https://mathoverflow.net/users/1231 | Is the space of solutions to the Queens Domination Problem connected? | Revised and correct:
589 equivalence classes.
<http://www.math.ucsd.edu/~etressle/classes.txt>
It seems that the pairs (#vertices,#components) are (1,388) (2,100) (3,40) (4,34) (5,20) (18,4) (20,2) (3804,1) – damiano 15 hours ago
```
1: Class I.D. 2, 4, 5, 6, 7,..., 589.
2: Class I.D. 22, 29, 35, 47, 48,..., 5... | 11 | https://mathoverflow.net/users/35336 | 36703 | 23,612 |
https://mathoverflow.net/questions/36708 | 12 | In singular (co)homology, if $\alpha\in C^\*(X)$ and $x\in C\_\*(X)$, then the cap product $\alpha \cap x$ is generally defined by the following process:
1. Apply to $x$ the diagonal map $C\_\*(X)\to C\_\*(X\times X)$ followed by some choice of Alexander-Whitney chain equivalence $ C\_\*(X\times X)\to C\_\*(X)\otimes... | https://mathoverflow.net/users/6646 | Conventions for definitions of the cap product | This is just an expanded version of Tyler's comment, I think.
Let's use a, b, c for cochains, x, y, z for chains, [a,x] for the value of a cochain on a chain. I'll be lazy and write $ab$ for $a\cup b$ and $ax$ for $a\cap x$. Let $[a\otimes b,y\otimes z]=(-1)^{|b||y|}[a,y][b,z]$.
I like to define $bx$ in such a way ... | 8 | https://mathoverflow.net/users/6666 | 36718 | 23,623 |
https://mathoverflow.net/questions/36714 | 3 | Is there a standard notation for a graph (on a given set of vertices) without any edges?
| https://mathoverflow.net/users/7732 | Notation for a graph without any edges? | There are many ways to define a *graph*, but a pretty standard one is a pair $(V,E)$ where $V$ is a finite set of points and $E \subset \binom{V}{2}$. So, what you are looking for is $(V, \emptyset)$; which would be pretty widely understood.
| 6 | https://mathoverflow.net/users/2233 | 36732 | 23,633 |
https://mathoverflow.net/questions/36729 | 3 | Consider the category of finite graphs with graph homomorphisms as morphisms.
>
> Are there interesting graph properties
> that can be defined in categorical
> language? Can for example
> connectedness be defined in
> categorical language?
>
>
>
| https://mathoverflow.net/users/2672 | Graph properties, categorically defined | Disjoint union is the coproduct in the category of finite graphs, so connected graphs are precisely the noninitial objects in this category that can not be expressed as a coproduct of two nonempty subobjects. See the entry on [connected object](http://ncatlab.org/nlab/show/connected+object) in the nlab.
If you want ... | 5 | https://mathoverflow.net/users/2384 | 36740 | 23,635 |
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