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https://mathoverflow.net/questions/115777 | 0 | I asked this via MathSE, but haven't got any responces. Sorry for asking it here. Sorry.
We know that in the context of *abelian* groups, $p$-groups are called $p$-primary groups. I have a question about $p$-primary groups as follows. Derek J.S.Robinson, noted:
>
> ...the group $\mathbb Q/\mathbb Z$ is the direct... | https://mathoverflow.net/users/13898 | $p$-primary then divisible? | Robinson only assserts that for the specific group $\mathbb{Q}/\mathbb{Z}$ the $p$-primary componenents/subgroups are divisible; and this would remain true replacing $\mathbb{Q}/\mathbb{Z}$ by any other divisible group.
Rotman's lemma is very different. Here, you suppose $G$ is divisible and in addition that it is $p... | 2 | https://mathoverflow.net/users/nan | 115786 | 65,514 |
https://mathoverflow.net/questions/115798 | 2 | I saw a statement in [Murakami, On automorphisms on Siegel domains] that every linear automorphism $\phi$ on the set of positive definite matrices can be represented as conjugation: i.e. there is a matrix $B\in GL(n,\mathbb{R})$ such that $\phi(A)=B^t A B$. It seems an easy statement but I couldn't prove it. Can somebo... | https://mathoverflow.net/users/6569 | On linear automorphism on positive definite matrices. | I assume that you mean 'positive definite *symmetric* matrices'. Here's one proof, though it's certainly not the most clean.
A linear automorphism of the space of all symmetric matrices that preserves the cone of positive definite matrices will preserve the closure of that cone, i.e., the *nonnegative* symmetric matr... | 4 | https://mathoverflow.net/users/13972 | 115808 | 65,524 |
https://mathoverflow.net/questions/115811 | 7 | What is the best reference in English for the following theorem of Grauert–Riemenschneider:
Theorem:
Let $\phi:X \to Y$ be a proper bi-rational morphism of algebraic varieties over characteristic $0$ field. Assume that $X$ is smooth. Let $\Omega\_X$ be the sheaf of top differential forms on $X$.Then $R\phi\_\*(\Omega... | https://mathoverflow.net/users/4690 | English reference for the Grauert–Riemenschneider vanishing theorem | Dear Rami,
You could see Kollar-Mori, *Birational geometry of algebraic varieties.* (Page 73)
or
Lazarsfeld, *Positivity in Algebraic Geometry I and II*. (Page 257)
| 11 | https://mathoverflow.net/users/3521 | 115814 | 65,527 |
https://mathoverflow.net/questions/115483 | 3 | Edited:
I guess
$$H^2\_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$
We know that if $\operatorname{Supp} H^i\_I(M)\subseteq V(I)\cap \operatorname{Supp}(M)$, then
$$\operatorname{Supp} H^2\_{(x,y)}\frac{\Bbb Z[x,y]}{(5x+4y)})\subseteq V((x,y))\cap V((5x+4y))=V((x,y))=\lbrace(x,y) \rbrace\cup \lbrace(... | https://mathoverflow.net/users/29695 | vanishing of local cohomology $H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$ | $P=(x,y,p)$ implies $P\cap\mathbb Z=p\mathbb Z$ (here $p$ is a prime or $0$). As localization commutes with local cohomology
$$H^2\_{(x,y)}\left(\frac{\mathbb{Z}[x,y]}{(5x+4y)}\right)\_P\simeq H^2\_{(x,y)}\left(\frac{\mathbb{Z}[x,y]\_P}{(5x+4y)}\right).$$
But $\mathbb Z[x,y]\_P\simeq\mathbb Z\_{(p)}[x,y]\_{\overline{P... | 2 | https://mathoverflow.net/users/23950 | 115817 | 65,529 |
https://mathoverflow.net/questions/115784 | 12 | I was reading through [Agol's paper on the Virtual Haken Conjecture](http://arxiv.org/abs/1204.2810) and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their stabilizers under the action of a hyperbolic group on a CAT(0) cube complex:
**Claim:** Su... | https://mathoverflow.net/users/29775 | Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite | Here's a proof.
**Lemma:** Suppose $G$ is a (word-)hyperbolic group acting properly discontinously, cocompactly and faithfully on a CAT(0) space $X$. Then there is a uniform bound $R\_0$ on the width $R$ of isometrically embedded flat strips $\mathbb{R}\times [0,R]$ in $X$.
*Proof:* If not then, by cocompactness... | 13 | https://mathoverflow.net/users/1463 | 115822 | 65,530 |
https://mathoverflow.net/questions/115809 | 9 | Consider the theory $\mathrm{PA}+\mathrm{BHO}$ consisting of first-order Peano arithmetic ($\mathrm{PA}$) enriched by an axiom scheme which allows well-founded induction up to any ordinal less than [a standard recursive presentation of] the Bachmann-Howard ordinal. More precisely, let $\prec$ be the explicit well-order... | https://mathoverflow.net/users/17064 | Arithmetic strength of Peano + the Howard ordinal | The answer is *yes*, using the ordinal analysis of KP. See Pohlers' *A Short Course in Ordinal Analysis* for why the usual ordinal analyses are *profound*, that is, they imply conservativity of the analyzed theory over PA + TI($\prec\_i$) for arithmetical sentences. Here TI($\prec\_i$) is the scheme of transfinite indu... | 9 | https://mathoverflow.net/users/2004 | 115826 | 65,531 |
https://mathoverflow.net/questions/115816 | 2 | Let $[n]$ denote the subset of $2n+1$ integers with absolut value at most $n$.
Consider $k$ (not necessarily different) vectors in $[n]^d$ summing up to zero. What is the smallest $k>1$ (for fixed $n$ and $d$) that assures the existence of a strict subsequence of vectors also summing up to zero?
I am most interested ... | https://mathoverflow.net/users/nan | Zero subsums of integer vectors | For n=1 we have recently solved the problem, the answer is around $d^d$, see <http://arxiv.org/abs/0912.0424> By we I mean my coauthors and by solve I mean gave a reasonable lower and upper bound.
| 3 | https://mathoverflow.net/users/955 | 115827 | 65,532 |
https://mathoverflow.net/questions/115810 | 1 | I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:
I have the homogeneous polynomial function
$$f(X) = \sum\_{u\_1, \dots u\_n = 1}^n X\_{u\_1} \cdots X\_{u\_n}$$
where $n$ is... | https://mathoverflow.net/users/16702 | Combinatorics: Product Rules. | It seems to me that in either your expression for $L$ or for $L^{n/2}f(X)$, you need to replace $\lambda\_j$ with $1/\lambda\_j$. (I see this has been corrected, so my previous sentence is irrelevant.) In any event, since $f(X)=(X\_1+\cdots+X\_n)^n$ it is clear that $C(n)=n!$.
| 5 | https://mathoverflow.net/users/2807 | 115828 | 65,533 |
https://mathoverflow.net/questions/115821 | 1 | There is a result in Cartan Eilenberg (XII 6.4) that says that if $G$ is a finite group and $D$ a divisible abelian group with trivial $G$-action then for any $G$-module $M$ the cup product $$\widehat{H}^{r-1}(G,Hom(M,D)) \times \widehat{H}^{-r}(G,M) \longrightarrow \widehat{H}^{-1}(G,D)$$ induces an isomorphism $$\wid... | https://mathoverflow.net/users/15566 | cap products and injective abelian groups | The isomorphism, at least, exists. Take $M\leftarrow Y$ and $\mathbb Z\leftarrow X$ projective resolutions of $M$ and of $\mathbb Z$ as $G$-modules. The double complex $\hom\_G(X,\hom\_{\mathbb Z}(Y,D))$ gives rise, as usual, to two spectral sequences. Computing first cohomology with respect to the differential induced... | 1 | https://mathoverflow.net/users/1409 | 115831 | 65,535 |
https://mathoverflow.net/questions/115823 | 6 | Let $U$ be a bounded domain in $R^2$ and let $n : U \to S^2$. Which (necessary/sufficient) conditions must $n$ satisfy in order that there exist an immersion $f : U \to R^3$ such that $n(x)$ is the normal to $f$ at the point $f(x)$, for all $x\in U$ ? The ideal answer would be of the form: "such $f$ exists if and only ... | https://mathoverflow.net/users/29785 | Conditions on a unit vector field to be the Gauss map of some surface immersed in R^3? | As Alexandre Eremenko points out, there's no PDE that $n$ would have to satisfy, at least for local solvability, which is believable when you think of it in heuristic terms: There are $3$ unknowns involved in specifying an immersion $f:U\to\mathbb{R}^3$ and only $2$ arbitrary functions needed to specify $n:U\to S^2$, s... | 5 | https://mathoverflow.net/users/13972 | 115833 | 65,537 |
https://mathoverflow.net/questions/115735 | 65 | In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and other such precise information which usually can only be attached to things that exist. Maybe someone familiar with the ... | https://mathoverflow.net/users/1409 | Groups that do not exist | There was a point during the history of the Classification when pursuers of sporadic groups distinguished the Baby Monster, the Middle Monster and the Super Monster. The first two actually turned out to exist (though the word "Middle" was dropped), but the third turned out to be a dud.
<http://www.neverendingbooks.o... | 34 | https://mathoverflow.net/users/12610 | 115844 | 65,542 |
https://mathoverflow.net/questions/115281 | 4 | Apologies - a better explanation than I started with - thanks to people for helping. It is obvious that there are many bad cases for rank - the problem is are there a reasonable number of good cases?
A rank for fgp modules (fgp = finitely generated projective, as appropriate) over a unital C\* algebra $A$ can be giv... | https://mathoverflow.net/users/29625 | rank of fin gen projective modules over C* algebras | Here is an answer to the second question (item (4) of the list in the question): The set of possible "matrix traces" will contain an invertible element if and only if the projection (or the module) is full. That is, the entries of the projection span the algebra as a two sided ideal in $A$. (This is also equivalent to ... | 4 | https://mathoverflow.net/users/13381 | 115848 | 65,545 |
https://mathoverflow.net/questions/115839 | 13 | As is well known, the space of solutions of a linear ODE with, say, $\mathcal{C}^\infty$ coefficients on $\mathbb{R}$ is a finite dimensional affine space (a vector space, in the homogeneus case).
>
> What is the structure of the space of solutions of a *non linear* ODE? In which cases does it have a "natural" stru... | https://mathoverflow.net/users/4721 | What is the structure of the space of solutions of a non linear ODE? | As Robert Bryant observed, something like the solubility of $F$ for $u^{(n)}$ needs to be assumed. Then by the trick (due I believe to D'Alembert) of setting $x=(t,u,\dots,u^{(n-1)})$ and dt/ds=1, the equation can always be rewritten
$$
\frac{dx}{ds}=f(x).
$$
This is what most geometers would call the "standard ODE", w... | 12 | https://mathoverflow.net/users/19276 | 115854 | 65,548 |
https://mathoverflow.net/questions/115825 | 8 | Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^\*)$). In examples, they always pick $V = H^1$. Can anyone give me an example where $V$ is chosen to be something other than $H^1$. Does it always need t... | https://mathoverflow.net/users/28178 | abstract evolution equations | Hille-Yosida existence theorem theorem is one example of abstract evolution equation. In its simplest form it starts with a closed, densely defined operator $A$ on a Hilbert space $H$ with domain $D(A)$ such that
$$(Au, u)\geq 0,\;\;\forall u\in D(A) $$
and $\boldsymbol{1}+ A: D(A)\to H$ is surjective. (Such operat... | 8 | https://mathoverflow.net/users/20302 | 115859 | 65,552 |
https://mathoverflow.net/questions/115628 | 3 | Suppose $P$ is a convex lattice polytope in $Z^3$ **without** interior lattice points, and we require the interior lattice points of each facet(i.e. dimensional 2 faces) are neither too much nor too few, say, they are less or equal to 2 but bigger or equal to 1.
Is there any classification results on such polytopes?... | https://mathoverflow.net/users/29730 | Classification of lattice polytopes with small number of lattice points in the facets | This is something you might be aware of already (and which does not seem to be exactly the situation you are formulating), but Reeve's tetrahedron is a 3-dimensional integral convex polytope without lattice-points in the interior. Furthermore, the only lattice points are the vertices, so there are four of them.
EDIT... | 3 | https://mathoverflow.net/users/11674 | 115865 | 65,554 |
https://mathoverflow.net/questions/93422 | 9 | Mathematically, completely positive maps on C\*-algebras generalize positive linear functionals in that every positive linear functional on a C\*-algebra $A$ is a completely positive map of $A$ into $\mathbb{C}$. Furthermore, we have the Stinespring construction as a powerful generalization of the GNS construction.
... | https://mathoverflow.net/users/6269 | What is the physical difference between states and unital completely positive maps? | The main interpretation, which is fundamental in quantum information theory, is that the transpose of a UCP map $E$ is a linear map on quantum states that represents a realistic information channel. This is the correct generalization of a Markov map or a stochastic map in classical probability theory. Such a map $E^T$ ... | 9 | https://mathoverflow.net/users/1450 | 115873 | 65,558 |
https://mathoverflow.net/questions/115838 | 7 | This question comes from <https://stackoverflow.com/questions/13747873/why-does-this-prime-function-work>, where somebody wrote a standard Sieve of Erathostenes algorithm --- with a bug. However, the bug does not have any bad effect up to 1'000'000. At this point, it would be interesting to know if we can find either a... | https://mathoverflow.net/users/29789 | Sieve of Erathostenes: removing consecutive items | There are more efficient ways to sieve, however the question as asked is interesting. I think that there will be a failure for $p=73$ at about $3.08 \times 10^{27}$. First a reformulation which I *think* is equivalent:
>
> Choose an odd prime $p$ and color the integers (starting at $2$) according to least prime div... | 8 | https://mathoverflow.net/users/8008 | 115879 | 65,560 |
https://mathoverflow.net/questions/115887 | 1 | Hi!
Let $(M,g)$ be a compact Riemannian manifold without boundary of dimension $2m$. Let
$$T:W^{2,2}(M)\rightarrow L^{2}(T^{\*}M\otimes TM)$$
be a second order, linear, differential operator (coefficients in local coordinates are bounded smooth functions).
Now consider the fourth order linear operator
$$\Lambda... | https://mathoverflow.net/users/4971 | Question about coercivity of a functional | I could not quite understand your question due to TeX problems and here is my best guess. There exists $C>0$ such that for any $f\in W^{4,2}$ orthogonal to $\ker \Lambda$ we have
$$C \Vert f\Vert\_{L^2} \leq \Vert \Lambda f\Vert\_{L^2}= \Vert Tf\Vert\_{L^2}^2.\tag{1}\label{1} $$
(For a proof of (\ref{1}) check Lemm... | 0 | https://mathoverflow.net/users/20302 | 115889 | 65,566 |
https://mathoverflow.net/questions/115898 | 4 | I am assuming that uniform spaces are Hausdorff (although it probably doesn't matter for this question). It is more-or-less obvious that a uniform space can be embedded in a product of metric space (if d is a semi-metric on the space) form the quotient gotten by identifying pairs of points at d-distance 0 and then map ... | https://mathoverflow.net/users/24338 | Does a uniform space have a closed embedding in a product of metric spaces? | You are looking for the notion of [Dieudonne complete](http://en.wikipedia.org/wiki/Completely_uniformizable_space) spaces (which turn out to be exactly the closed subspaces of products of metric spaces). As Todd mentioned in his comment, this notion is closely related with the notion of realcompactness (the two notion... | 5 | https://mathoverflow.net/users/17836 | 115906 | 65,571 |
https://mathoverflow.net/questions/115891 | 0 | Let $f$ be an entire function of order $ρ<\infty$. Assume that $f$ does not vanish identically on $\mathbb{C}$. Then, we know that $f$ has a Hadamard's product formula
$$ f(s) =e^{g(s)}s^{r}\prod \_ {k=1}^{\infty}\frac{s \_ {k}-s}{s \_ {k}} e^{s/s \_ k} $$
the integer $r$ is the order of vanishing of $f$ at $s=0$, ... | https://mathoverflow.net/users/25947 | Hadamard's product formula for the derivative | (1) It seems your formula for the Hadamard product is only correct for $\rho<2$; more generally the exponent of $e^{s/s\_k}$ contains a power series in $s$ of order $q={\rm Int}\;\rho$; see for example Eq. 1 in these [lecture notes](http://www.math.harvard.edu/~elkies/M259.06/prod.pdf).
(2) To find a similar expressi... | 2 | https://mathoverflow.net/users/11260 | 115907 | 65,572 |
https://mathoverflow.net/questions/115895 | 6 | I want to ask the following probability inequality:
Is it true that for any random variable $X\ge 0$, we have
$$
\sup\_{t>0}(t\mathbb E(X\mathbf 1\_{X\ge t}))
\le
2\sup\_{t>0}(t^2 \mathbb P(X \ge t))?
$$
| https://mathoverflow.net/users/12264 | Inequality involving the weak second moment | This is an interesting inequality.
Let $f$ be the density of $|X|$.
Let
$$g(t)=t\int\_t^\infty xf(x)dx.$$
Then our inequality is
$$\sup\_t g(t)\leq \sup\_t t^2\int\_t^\infty f(x)dx.$$
Let us assume that $\sup$ of the LHS is attained at some point $t=a$,
so that $g(a)$ be the maximal value of $g$.
(It will be easy ... | 6 | https://mathoverflow.net/users/25510 | 115915 | 65,575 |
https://mathoverflow.net/questions/115866 | 11 | I am a physics student, recently I read a paper using Homotopy $\pi\_4(SU(2))=Z\_2$, I guess mathematicians have some **visualization or explanation** of this result. So I come here ask for help.
CROSS-POST from <https://physics.stackexchange.com/questions/46284/homotopy-pi-4su2-z-2>
| https://mathoverflow.net/users/24094 | Homotopy $\pi_4(SU(2))=Z_2$ | This calculation of $\pi\_4(S^3)$ is also obtained in the paper R. Brown and J.-L. Loday, Topology, 26 (1987) 311-334, and also available [here](http://groupoids.org.uk/pdffiles/RB-Loday1.pdf). In that paper, $S^3$ is regarded as the double suspension $SS$ of the circle $S^1$, which is itself seen as an Eilenberg-Mac L... | 11 | https://mathoverflow.net/users/19949 | 115931 | 65,585 |
https://mathoverflow.net/questions/115875 | 13 | My favorite model of quantum probability is by von Neumann algebras, i.e., a quantum measurable space is a von Neumann algebra and a quantum distribution is a normal state. Then, one important new phenomenon in quantum probability is the existence of bosons and fermions. Another important new phenomenon, of which boson... | https://mathoverflow.net/users/1450 | Which von Neumann algebras have inner permutation of tensor factors? | Here's an argument showing that in the ${\rm II}\_1$ case the flip automorphism is never inner.
Let $M$ be a type ${\rm II}\_1$ factor and $\tau$ its trace, so that $M \subset L^2(M, \tau)$, and $M$ acts standardly on $L^2(M, \tau)$. Suppose that the flip automorphism is implemented by a unitary $U \in \mathcal U(M ... | 18 | https://mathoverflow.net/users/6460 | 115936 | 65,588 |
https://mathoverflow.net/questions/115939 | 0 | For independent Rademacher random variables $\epsilon\_i, i=1,2, \cdots, n$, i.e. $P(\epsilon\_i=-1)=P(\epsilon\_i=1)=\frac{1}{2}$, do we have
$$max\_{0\le a\_i\le b, i=1,2,\cdots,n}P(|\sum\_{i=1}^n\epsilon\_ia\_i^2|>x)\le max\_{0\le a\_i\le b, i=1,2,\cdots,n}P(|\sum\_{i=1}^n\epsilon\_ia\_i|>x), \forall x>0,$$
where... | https://mathoverflow.net/users/12264 | a simple probability inequality | No. Let $a\_1 = 1, a\_2 = 1/2, x=2/3$.
$P(|\epsilon\_1 + 1/4 \epsilon\_2| > 2/3) = 1$
$P(|\epsilon\_1 + 1/2 \epsilon\_2| > 2/3) = 1/2$
| 2 | https://mathoverflow.net/users/2954 | 115941 | 65,589 |
https://mathoverflow.net/questions/115908 | 8 | I learned [this question](https://math.stackexchange.com/questions/235689/pointwise-existence-of-radon-nikodym-derivative-sufficient-for-absolute-continui) from math.stackexchange, which is equivalent to ask that if $f:[0,1]\to \mathbb{R}$ is a continuous function with bounded variation, does
$$g(x):=\lim\_{\epsilon\t... | https://mathoverflow.net/users/29804 | Does a weaker condition than vanishing derivative imply a function being constant? | The ordinary proof that 0 derivative implies constant rests on the mean value theorem. Your function $g(x)$ is the "symmetric derivative" of $f$, so we should look for a "quasi-mean value theorem" for symmetric derivatives. A quick google search came up with the paper
<http://www1.au.edu.tw/ox_view/edu/tojms/j_paper... | 10 | https://mathoverflow.net/users/7399 | 115944 | 65,592 |
https://mathoverflow.net/questions/115945 | 1 | Let $R$ be a commutative unitary ring and $M\_{I}$ be the intesection of all maximal ideals contains $I$.
Question: When for any two ideals $I$ and $J$ of $R$ there exists an ideal $K$ of $R$ such that $M\_{I}+M\_{J}=M\_{K}$?
| https://mathoverflow.net/users/29349 | Intersection of maximal ideals contains an ideal | A typical ring of dimension $2$, like $k[x,y]$, does not have that property. Indeed, if $I=(xy)$, $J=(x^2-y^2)$, then $M\_I=I$, $M\_J=J$, and $M\_I+M\_J=I+J$ contains $x^3$ and $y^3$ but not $x$ and $y$, so is not radical, so cannot be the intersection of any set of maximal ideals.
On the other hand, in a Dedekind do... | 2 | https://mathoverflow.net/users/18060 | 115946 | 65,593 |
https://mathoverflow.net/questions/115953 | 3 | I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime distinct from the base field characteristic (the latter could be $0$). I would like to have a characterization of this van... | https://mathoverflow.net/users/2191 | $Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$? | $Pic(X)$ mod $l$ injects into $H^2(X,\mu\_l)$ with cokernel the group of elements of order $l$ in the Brauer group of $X$. Depending on your $X$, the Brauer group may be known, or it may be as mysterious as a Tate-Shafarevich group. For example, for a complete smooth surface over a finite field, the Brauer group is con... | 5 | https://mathoverflow.net/users/22295 | 115957 | 65,597 |
https://mathoverflow.net/questions/47836 | 25 | Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V$ the subbundle of $E=TM\oplus T^\*M$ given by the graph of the musical linear isomorphism $g^\flat:TM\rightarrow T^\*M$ associated to the metric $g$. The nondegeneracy of $g$ entails that $E$ decomposes as the Whitney sum $E=V\oplus V'$, wher... | https://mathoverflow.net/users/11211 | Generalized geometry and spin structures | OK, I am making an assumption: *I can re-interpret the problem (using the musical isomorphism) as $V$ being the diagonal embedding of $TM$ inside $TM\oplus TM$ and studying spin structures on them*.
**Actually, it turns out** *(see comments)* **that this "re-interpretation" is slightly different from the original con... | 4 | https://mathoverflow.net/users/12310 | 115964 | 65,600 |
https://mathoverflow.net/questions/115940 | 1 | That is: is it true that if projective k[G]-modules have same composition factors then they are isomorphic?
This is easy to see for char(k)=0, or if G is a composition of a p-group and a p′-group. Serre in "Linear Representations of Finite Groups" (a remark in 16.2 after Corr.2) states this as a well-known fact: "Ind... | https://mathoverflow.net/users/29813 | For finite group G and field k of char=p, if P,P′ are projective k[G]-modules with [P]=[P′], is it true that P=P′ ? | Some clarifications to the question are needed. First, you are referring to Section 16.1 of Serre's book (not 16.2), where he is formulating the main results in Brauer theory. These were originally derived (as in the 1962 Curtis-Reiner text) more concretely in terms of Brauer characters, but then recast in the language... | 4 | https://mathoverflow.net/users/4231 | 115975 | 65,605 |
https://mathoverflow.net/questions/103949 | 37 | Suppose that $\gamma$ is a Jordan analytic curve on the Riemann sphere,
and there exist two rational functions $f$ and $g$ such that
$f$ maps $\gamma$ into a circle, and $g$ maps a circle into $\gamma$.
(All rational functions considered are of degree at least $2$).
Question: Does this imply that $\gamma$ is a circle... | https://mathoverflow.net/users/25510 | Circles and rational functions | **Update:** I removed the links to the Sage code of the complicated explicit examples, because very much easier examples exist. See below for an example, and [this preprint](http://arxiv.org/abs/1502.07336) for more details concerning this answer and the computation of explicit examples.
**Answer:** The answer is no.... | 26 | https://mathoverflow.net/users/18739 | 115979 | 65,608 |
https://mathoverflow.net/questions/115958 | 14 | Consider the Euclidian space $E\_n={\mathbb R}^n$, with standard scalar product
$$x\cdot y=x\_1y\_1+\cdots+x\_ny\_n.$$
A closed convex cone $\Gamma\subset E\_n$ defines an order by $y\ge x$ iff $y-x\in\Gamma$. An order is *compatible* with the Euclidian structure if
* $x,y\in\Gamma$ implies $x\cdot y\ge0$,
* convers... | https://mathoverflow.net/users/8799 | Convex cones and self-duality | For convex figure $\Sigma$ in $\mathbb S^2$,
the isoperimetrical inequality should look like
$$\left(\frac{\mathop{\rm perim}\Sigma}{2\cdot\pi}\right)^2+\left(1-\frac{\mathop{\rm area}\Sigma}{2\cdot\pi}\right)^2\ge 1.$$
If $\Sigma$ and $\Sigma'$ are the intersections of $\mathbb S^2$ with your cones
then by Crofton f... | 10 | https://mathoverflow.net/users/1441 | 115985 | 65,610 |
https://mathoverflow.net/questions/115966 | 5 | Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map
$\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$.
Can you describe the branch locus of $\phi$ in terms of Schubert classes?
| https://mathoverflow.net/users/27125 | Branch locus of a 6:1 cover of the grassmannian G(1,3) | Since you are interested in a divisor, you only need to know its degree, that is its intersection with a line. A generic line on $Gr(1,3)$ is given by the set of all lines contained in a plane $P$ and passing through a point $Q$. So, you want to know how many tangents to $S$ pass through $Q$ and lie in $P$.
Consider... | 12 | https://mathoverflow.net/users/4428 | 115993 | 65,612 |
https://mathoverflow.net/questions/115991 | 6 | This question comes from a problem in PDEs on which I'm currently working. Let $a$ be a $3\times 3$ matrix, real symmetric and positive definite. Denote with $\|a\|^2 \_ 2=\sum a\_{ij}^2$ the square of the Hilbert-Schmidt norm and consider the quantity
$$ Q(v)= 2\|a\|^2 \_2 + [trace(a)-3(av,v)]^2
-6[2 |av|^2 -(av,v)^2... | https://mathoverflow.net/users/7294 | A matrix inequality involving the Hilbert-Schmidt norm | Suppose $Q$ is such a form. Write that the mean value of $Q$ over the unit sphere is non-negative. You obtain
$$-\frac12\sum\_ia\_{ii}^2+\frac12\sum\_{i < j}a\_{ii}a\_{jj}-9\sum\_{i < j}a\_{ij}^2\ge0.$$
This implies that $a=\lambda I\_3$.
| 8 | https://mathoverflow.net/users/8799 | 115994 | 65,613 |
https://mathoverflow.net/questions/115997 | -1 | I am dealing with an integral in a limit of the following shape:
$$\lim\_{\epsilon \to 0} \int\_0^{\frac{\pi}{2}} dx \frac{2 \epsilon}{1-(1-\epsilon^2)\sin^2(x)}$$
Formally, assuming that $x=\arcsin(\frac{1}{\sqrt{1-\epsilon^2}})$ is a possible value on the integration path, we obtain a non zero residue (independen... | https://mathoverflow.net/users/26176 | Residue at an integration border in case of a limit? | Of course, you cannot put $\epsilon=0$ under the integral.
You must evaluate it for $\epsilon\neq 0$ and then pass to the limit.
The correct way to find this integral is to spread it from $-\pi/2$ to $\pi/2$,
using that your function is even, and then make the change of the variable
$z=\exp(ix)$. This will reduce to ... | 1 | https://mathoverflow.net/users/25510 | 116001 | 65,616 |
https://mathoverflow.net/questions/115438 | 15 | Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. Let me try again, hoping that the questions make more sense now.
Second Edit: I have added a bounty to the question. I ... | https://mathoverflow.net/users/9317 | Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ? | The basic heuristic behind the scenes in all of these conjectures is that there is optimal "square-root" cancellation in the various error terms. The estimate (2) is surely not true; the explicit formula links this to a sum over zeros of the related Dirichlet L-function, and it has zeros on the half-line. Instead, one ... | 5 | https://mathoverflow.net/users/2627 | 116012 | 65,619 |
https://mathoverflow.net/questions/116003 | 10 | For certain values of $k$ it is known that $\mbox{Diff}(S^k)$ is not homotopy equivalent to $O(k+1)$. So there are sphere bundles that do not arise from vector bundles.
Since I've never (knowingly) come across such a sphere bundle I'm interested in seeing some enlightening examples of sphere bundles which do not come... | https://mathoverflow.net/users/29827 | Examples of sphere bundles | As far as I know the only explicitly-described such bundles are in Hatcher's paper:
* Hatcher. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ, Stanford Calif 1976), Part 1, pp. 321.
I think in Igusa's Higher Franz Reidem... | 5 | https://mathoverflow.net/users/1465 | 116013 | 65,620 |
https://mathoverflow.net/questions/115996 | 1 | Hi all. My question on M.SE is unanswered (<https://math.stackexchange.com/questions/254970/fourier-transform-of-function-defined-on-subset-of-mathbbrn>) so I want to post it here. I changed it slightly.
If I have a function $f:\Omega \to \mathbb{R}$ in $H^k(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is a compact s... | https://mathoverflow.net/users/29829 | Fourier transform of function on compact set and Sobolev norm equivalence | This seems to be one special case of the so called *Fourier restriction problem* (see, for instance, Terry Tao's discussion: <http://terrytao.wordpress.com/2010/12/28/the-bourgain-guth-argument-for-proving-restriction-theorems/>). If $\Omega$ is a smooth, $(n-1)$-dimensional embedded submanifold (compact or not) of $\m... | 2 | https://mathoverflow.net/users/11211 | 116014 | 65,621 |
https://mathoverflow.net/questions/115972 | 1 | Hello,
Let $x\in[0;1]$ and $(B\_i)\_i$ be events defined by $P(B\_i)\leq x, \forall i$.
Furthermore, this inequality is independent of the other events $B\_i$ but the events are not necessarily independent.
I want to upperbound the probability of $A\_k = (B\_1\cup B\_2)\cap (B\_2\cup B\_3)\cap\cdots\cap (B\_k\cup B... | https://mathoverflow.net/users/22976 | Apparently simple probability | Let $m:=\big\lfloor\frac{k+1}{2} \big \rfloor$. Then, assuming $\mathbb{P}(B \_ i)\le x$,
$$\mathbb{P}\big((B \_ 1\cap B \_ 2) \cup(B \_ 2\cap B \_ 3) \dots \cup(B\_k\cap B\_{k+1})\big)\le mx\wedge 1\, ,$$
which follows immediately by the inclusion
$$(B \_ 1\cap B \_ 2) \cup(B \_ 2\cap B \_ 3)\cup \dots \cup(B\_k\cap... | 1 | https://mathoverflow.net/users/6101 | 116021 | 65,626 |
https://mathoverflow.net/questions/116022 | 13 | (This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.)
[This paper](http://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0251026-X/S0002-9947-1969-0251026-X.pdf) gives a proof that the underlying topological spaces of af... | https://mathoverflow.net/users/25342 | Which local ringed spaces are schemes? | I think the easiest condition is the fact that the natural morphism
$$
(X, \mathcal O\_X) \to \operatorname{Spec}(\mathcal O\_X(X))
$$
is an isomorphism.
| 16 | https://mathoverflow.net/users/7845 | 116029 | 65,628 |
https://mathoverflow.net/questions/116026 | 1 | How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
| https://mathoverflow.net/users/29837 | Calculating the Lebesgue decomposition of a measure | Let e.g. $\nu$ be a finite Borel measure on $[a,b]$ and $m$ the Lebesgue measure. So the function $[a,b]\ni x\mapsto \nu\big([a,x)\big)$ is a BV function. It is therefore differentiable $m$-a.e., and its derivative $\rho(x)$ coincides with the Radon-Nikodym derivative of the absolutely continuous part of $\nu$. Knowing... | 1 | https://mathoverflow.net/users/6101 | 116030 | 65,629 |
https://mathoverflow.net/questions/116034 | 4 | When considering classification problems about polytopes, I sometimes has the feeling that one need to talk about certain parametrized families, i.e. moduli space of such polytopes. But neither do I have a concrete example on hand nor do I know how to formulate the definition of such moduli space. Does anyone know the ... | https://mathoverflow.net/users/29730 | moduli space of polytopes | One thing that is commonly done is to fix an initial polytope $P$, and consider all the polytopes whose fans are coarsenings of $P$'s fan. You can parametrize these by the space of convex piecewise-linear functions on $P$'s fan, to see that the moduli space itself forms a polyhedral cone.
This is no good if you want ... | 5 | https://mathoverflow.net/users/391 | 116036 | 65,631 |
https://mathoverflow.net/questions/116037 | 1 | I would warmly appreciate it if someone could tell me whether the following question has an affirmative answer. I am new to the field of commutative algebra, so I am simply trying to fill in some (huge) gaps. Thanks!
Let $ (R,{\frak{m}}) $ be a Noetherian local (commutative unital) ring. Let $ I $ be an ideal of $ R ... | https://mathoverflow.net/users/26077 | A Question About Free Resolutions | No. Consider $\mathfrak{m}:=(x,y,z)\subset k[x,y,z]\_{(x,y,z)}=:R$. Then the kernel of the map $$R^3\to \mathfrak{m}$$ defined by the minimal generating set $x,y,z$ is minimally generated by $$k\_1:=(y, -x, 0), k\_2:=(z, 0, -x), k\_3:=(0, z, -y).$$ But $$zk\_1-yk\_2-xk\_3=0$$
so the submodule of $R^3$ that these genera... | 1 | https://mathoverflow.net/users/6950 | 116038 | 65,632 |
https://mathoverflow.net/questions/116042 | 4 | Suppose I have an $n\times n$ real (or complex) matrix of rank $k$, and I want to pick $k$ linearly independent rows from it. I want to do this in a continuous fashion as the matrix varies continuously. I'm being a bit vague here, but I think it doesn't matter, because a colleague tells me that it's a standard fact tha... | https://mathoverflow.net/users/3106 | Impossibility of continuously picking k independent rows from a rank k matrix | Picking $k$ linearly independent rows is harder than picking a basis for the row space. The row space forms a vector bundle on the manifold of rank $k$ matrices. Picking a basis continuously would be equivalent to picking a trivialization of the vector bundle.
So your colleague's claim is weaker than the fact that t... | 7 | https://mathoverflow.net/users/18060 | 116043 | 65,633 |
https://mathoverflow.net/questions/116045 | 4 | I've been reading Waterhouse's book "Introduction to affine group schemes", in part to help prepare myself for an (oral) advanced topic exam in algebraic geometry. There is one exercise in chapter 1 that has been giving me trouble. Let $G$ be an affine group scheme with associated Hopf algebra $A$. The exercise says th... | https://mathoverflow.net/users/29842 | Basic question about affine group schemes | You are confusing algebra isomorphisms with Hopf algebra isomorphisms. The map $A\otimes A \to A\otimes A$ given by $a\otimes b\mapsto \left(a\otimes 1\right)\left(\Delta\left(b\right)\right)$ is an algebra isomorphism but not a Hopf algebra isomorphism in general. So it corresponds not to a group automorphism of $G\ti... | 10 | https://mathoverflow.net/users/2530 | 116047 | 65,636 |
https://mathoverflow.net/questions/116048 | 2 | Pillai's conjecture -- that the gap between (nontrivial) powers is unbounded below -- is still open (it would be a consequence of the $abc$ conjecture, were that proven). But I wonder what the right order of magnitude for it is, even though a proof seems far off.
Suppose $n=|a^x-b^y|$ for integers $a,b,x,y$ with $a,b... | https://mathoverflow.net/users/6043 | Maximum size of powers with a given difference | [Perfect Powers: Pillai's works and their developments](http://arxiv.org/abs/0908.4031)
is related to your question.
p.9:
Conjecture 3.1. For any $\epsilon > 0$, there exists a constant $\kappa(\epsilon) > 0$ such that, for any positive integers $(a, b, x, y)$, with $x \ge 2, y \ge 2$ and $a^x \ne b^y$ ,
$$|a^x - b^y... | 4 | https://mathoverflow.net/users/12481 | 116050 | 65,638 |
https://mathoverflow.net/questions/116052 | 2 | Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as
$X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product).
Now the composition $M\circ E$ ($M$ followed by $E$) is a monad again.
At least I believe it is true; and I wonder if it is ... | https://mathoverflow.net/users/20031 | Seems like Reader monad composed with a strong monad produces a monad, am I right? | Although I believe the question needs cleaning up, what I believe to be the desired statement is true. I cannot find a direct reference for it at the moment, but the statement should be equivalent to something familiar from the literature.
First, the "Reader monad" for a fixed object $E$ crucially uses cartesian (not... | 7 | https://mathoverflow.net/users/2926 | 116065 | 65,644 |
https://mathoverflow.net/questions/116060 | 0 | Let
$$
A\_1\twoheadrightarrow
A\_2\twoheadrightarrow
A\_3\twoheadrightarrow
A\_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of **countable** abelian groups, the connecting homomorphisms of which are surjective and **split**, that is, we have embeddings $A\_{n+1}\rightarrowtail A\_n$ such that the composition ... | https://mathoverflow.net/users/1291 | Inductive vs projective limit of sequence of split surjections | I think it's true that $\varprojlim A\_n\to\varinjlim A\_n$ is always injective.
We may as well assume that
$A\_1\leftarrowtail
A\_2\leftarrowtail
A\_3\leftarrowtail
A\_4\leftarrowtail
\cdots$
is a sequence of inclusions of nested subgroups, so $\varprojlim A\_n$ is just the intersection. An element of the kerne... | 2 | https://mathoverflow.net/users/22989 | 116068 | 65,645 |
https://mathoverflow.net/questions/116067 | 3 | Let $E$ be a vector bundle on a smooth projective variety $X$ and assume $\mathrm{rk}(E)>\mathrm{dim}(X)$. If $E$ is globally generated, then a general section of $E$ is nowhere vanishing (see Question [Vanishing locus of a general section of a vector bundle.](https://mathoverflow.net/questions/63909/vanishing-locus-of... | https://mathoverflow.net/users/33841 | Generically generated vector bundles & degeneracy loci. | It seems to me that the answer is *no*, because of the following counterexample.
Take a surface $X$ with a linear pencil $|L|$ having a unique base point $x$. Now set $$E=\mathcal{O}\_X(L) \oplus \mathcal{O}\_X(L) \oplus \mathcal{O}\_X(L).$$
Since $L$ is generically globally generated, the same is true for $E$.
On ... | 2 | https://mathoverflow.net/users/7460 | 116072 | 65,647 |
https://mathoverflow.net/questions/116069 | 6 | There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms
in Real Algebraic Geometry) see <http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html>. This is a package for Maxima.
I looked into this and came across the Thom encoding of real algebraic numbers. If I have un... | https://mathoverflow.net/users/3992 | Exact arithmetic for real algebraic numbers | In the [reference](http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html) you posted there is a link to an online version of the monograph by S.Basu, M.-F.Roy, and R.Pollack, where algorithms like this are described (cf. Sect. 10.4, Algorithm 10.15).
The technique there is very general, and applies t... | 6 | https://mathoverflow.net/users/11100 | 116077 | 65,650 |
https://mathoverflow.net/questions/116078 | 0 | In the game of chess,it is a proven fact that either of the three conditions hold:
1)white has a winning strategy
2) black has a winning strategy or
3)either of them can at least force a draw.
It is conjectured that white has a winning strategy.
What is the evidence behind this conjecture (why is 1) favoured wrt the ot... | https://mathoverflow.net/users/23786 | a question on game of chess | It is important to set aside two different discussions. One is what happens with optimal play, and the other is what happens in real games among the chess elite players. Statistics for any period of time show more white wins than black. It is generally accepted that having white is an advantage, even if it is only psyc... | 3 | https://mathoverflow.net/users/13923 | 116084 | 65,653 |
https://mathoverflow.net/questions/116082 | 19 | I am reading "Category Theory" (2nd ed.) of Awodey, and I'm stuck at page 96 (proposition 5.12) when pullbacks are presented as functors:
The pullback under question corresponds to this square:
$$\begin{matrix}
C' \times\_C A & \xrightarrow{h'} & A \\[1ex]
\downarrow \rlap{\scriptstyle{\alpha'}} & & \downarrow\rla... | https://mathoverflow.net/users/29853 | Why pullback only defined up-to-isomorphism but nevertheless presented as functor? | Let us first consider a slightly simpler situation. The cartesian product of sets $A$ and $B$ is a set $C$ with two maps $p\_1 : C \to A$ and $p\_2 : C \to B$ such that ... (familiar condition inserted here). All cartesian products of $A$ and $B$ are canonically isomorphic, and among them there is a particular one, den... | 18 | https://mathoverflow.net/users/1176 | 116085 | 65,654 |
https://mathoverflow.net/questions/115980 | 3 | Let $i : Y \to X$ be a quasi-compact immersion of schemes and let $M$ be a quasi-coherent sheaf on $X$. There is a canonical homomorphism
$M \otimes i\_\* \mathcal{O}\_Y \to i\_\* i^\* M.$
**Question**: Is it always an isomorphism?
Clearly this question is local on $X$. The class of $M$ satisfying the condition i... | https://mathoverflow.net/users/2841 | Projection formula for immersions | Set $X = \mathbb A^2\_k$, $Y = X \smallsetminus \{(0,0)\}$, and suppose that $M$ is non-zero and supported at the origin.
| 3 | https://mathoverflow.net/users/4790 | 116098 | 65,658 |
https://mathoverflow.net/questions/115923 | 5 | In [one of their papers](http://www.ams.org/journals/tran/1994-342-02/S0002-9947-1994-1142778-X/S0002-9947-1994-1142778-X.pdf) (before Theorem 7.2), Benson and Carlson state that the transfer map is Tate-dual to the restriction homomorphisms (also see Remark 1.3 of this [recent paper](http://arxiv.org/pdf/1211.5999v1.p... | https://mathoverflow.net/users/27895 | Why is the transfer map Tate-dual to restriction ? | I don't know of a reference, but the duality in question can be proved by results from Brown's book on group cohomology. I'll show the case $s\ge 0$. First note that for each integer $j$ there is an isomorphism
$$\psi: \hat{H}^j(G,k) \xrightarrow{\sim} \text{Hom}\_k(\hat{H}\_j(G,k),k)$$
(Brown, VI.7.2) and for $s\... | 4 | https://mathoverflow.net/users/10194 | 116103 | 65,661 |
https://mathoverflow.net/questions/116106 | 3 | Let $f(t)$ and $g(t)$ be periodic functions on $t\in[0,2\pi]$. By using the Fourier series of the two functions, we can easily prove the inequality
$$\left|\int\_0^{2\pi}f(t)g'(t)dt\right|=
\left|\int\_0^{2\pi}f'(t)g(t)dt\right|\le
\frac{1}{2}\int\_0^{2\pi}[f'(t)^2+g'(t)^2]dt\text.$$
I have been trying to find a re... | https://mathoverflow.net/users/20186 | A Wirtinger-like inequality involving two functions | Your inequality is implicit in Hurwitz's Fourier series proof of the isoperimetric inequality in the plane. See for example section 36 of Körner's *Fourier Analysis* or section 4.1 of Groemer's *Geometric Applications of Fourier Series and Spherical Harmonics*.
| 6 | https://mathoverflow.net/users/1044 | 116108 | 65,662 |
https://mathoverflow.net/questions/116107 | 1 | Suppose $\delta\in (0,1)$ and $r<1+\delta.$ Suppose moreover we are given a sequence of functions $u\_m\in H^{1/2,2}(\partial B\_r(0))$, where $B\_r(0)$ denotes the euclidean $n-$dimensional ball. Assume that $u\_m\to 0$ strongly in $H^{1/2,2}(\partial B\_r(0))$. Then I would like to extend these functions to functions... | https://mathoverflow.net/users/nan | Continuity of an extension map | Yes, this extension works. Your extended function is clearly in $H^1$ on the annulus as well as
in $R^n\backslash B\_2$. Since the traces on $\partial B\_2$ agree, it is then in $H^1$ on $R^n\backslash B\_r$.
| 1 | https://mathoverflow.net/users/12120 | 116111 | 65,663 |
https://mathoverflow.net/questions/116112 | 0 | Hello.
I have a research note coming out soon, and I'm stuck showing that a weird kind of function is continuous. I need to it show a new method of bounding exponential growth factors in combinatorial classes.
The function in question is $f: [0,1] \rightarrow (0,\infty)$, which sends a parameter $l \in [0,1]$ to th... | https://mathoverflow.net/users/29860 | Continuity of critical points with respect to a parameterisation. | If you know that P''>0, then the implicit function theorem should be applicable to give you continuity.
| 2 | https://mathoverflow.net/users/12120 | 116113 | 65,664 |
https://mathoverflow.net/questions/116081 | 9 | Suppose $M$ is a complete Riemannian manifold with very large injectivity radius (say larger than $100$) and $\left\lbrace x\_i: i \in I\right\rbrace$ is a maximal $1$-separated subset of $M$.
Is diffeomorphism class of $M$ determined by the (possibly infinite) distance matrix $(d(x\_i,x\_j))\_{i,j \in I}$?
Suppose... | https://mathoverflow.net/users/7631 | Discretization of a complete manifold | My knowledge of this subject is obsolete, but anyway here are some partial answers.
If you have a Ricci curvature bound in addition to the injectivity radius, then you can recover the diffeomorphism type, e.g. with harmonic coordinates, see M. Anderson, Convergence and rigidity of manifolds under Ricci curvature boun... | 12 | https://mathoverflow.net/users/4354 | 116114 | 65,665 |
https://mathoverflow.net/questions/115978 | 9 | Hi friends,
I am looking for examples of curves over a number field such that their jacobians are CM abelian varieties by a field whose Galois group is non-abelian. Does anybody know how to produce such examples out of a curve with the action of a finite non-abelian group G?
Thanks a lot!
| https://mathoverflow.net/users/29826 | jacobians with non-abelian complex multiplication | I think a large group of automorphisms of a curve $C$ only forces
presence of roots of unity in the endomorphism algebra
$\text{End}^0(\text{Jac}\ C)$, so this way is likely to produce
products of abelian varieties with CM by abelian fields. There is a lot
of literature on such examples, e.g. on Fermat curves $x^n+y... | 19 | https://mathoverflow.net/users/3132 | 116115 | 65,666 |
https://mathoverflow.net/questions/116091 | 0 | I am discovering random graph and I am trying to prove the following result. This is a follow-up on a previous question of mine
[what's an upper bound on the size of the largest biclique in random bipartite graph?](https://mathoverflow.net/questions/115748/whats-an-upper-bound-on-the-size-of-the-largest-biclique-in-r... | https://mathoverflow.net/users/29764 | expected size of unbalanced biclique in random bipartite graph | Thanks for the clarification. This set-up has enough flexibility that we can recover the star counterexample from before.
Almost every random graph has a vertex in each class of degree about $pn$. So we can almost always choose $E$ to be a large star, and we can ensure the centre of the star is in each class about eq... | 0 | https://mathoverflow.net/users/25485 | 116128 | 65,670 |
https://mathoverflow.net/questions/116104 | 5 | This is a question about the proof of proposition 1.13 in Deligne and Milne, Tannakian Categories. Let $C,C'$ be two rigid tensor categories and $F,G : C \rightarrow C'$ be two tensor functors. Let $u : F \rightarrow G$ be a morphism of functors. Define the morphism $v : G \rightarrow F$ by
$$ v(X) : G(X) \simeq G(X^\v... | https://mathoverflow.net/users/10427 | Functors on rigid tensor categories. | Okay, here is a [link](http://ncatlab.org/toddtrimble/published/Morphisms+between+tensor+functors) to my web at the nLab which provides a diagrammatic proof (for one of the two equations that must be verified; the other equation is established similarly).
Of course, the gigantic diagram which you will find did not s... | 7 | https://mathoverflow.net/users/2926 | 116130 | 65,672 |
https://mathoverflow.net/questions/116118 | 3 | There is no entire function of order 1 and type 0 which is bounded on the real line. This result can for example be found in the book "Entire functions" by Boas.
I wonder under which weaker conditions of boundedness this is still true. For example, do there exist entire functions $f$ of order 1 and type 0, which in a... | https://mathoverflow.net/users/23753 | Entire functions of order 1 and type 0 | No, such functions do not exist. Let $u=\log|f|$. Your condition implies that $u(x)\leq O(|x|^\rho)$,
where $0\leq \rho<1$, so the Poisson integral of $u^+$ is convergent,
and one can obtain the formula
$$u(z)=\frac{y}{\pi}\int\_{-\infty}^\infty u(t)\frac{dt}{(t-x)^2+y^2}\quad\quad +ky+\log|B(z
)|,$$
where $z=x+iy$, an... | 8 | https://mathoverflow.net/users/25510 | 116136 | 65,674 |
https://mathoverflow.net/questions/116101 | 1 | The motivation for this question is the same as in my previous question in MO: <https://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over>
I am just curious to know the origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation $f(s)=w(C/ℚ)f(2-s)=εf(2-s)$) of the cur... | https://mathoverflow.net/users/25947 | The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation) | I am not sure I understand your question. The functional equation $f(s)=\epsilon f(1-s)$ applied twice yields $\epsilon^2=1$, hence $\epsilon=\pm 1$.
Now, finding the root number $\epsilon$ of a self-dual modular $L$-function can be a tricky business, and the meaning of "finding" is of course subjective. It is known ... | 5 | https://mathoverflow.net/users/11919 | 116152 | 65,683 |
https://mathoverflow.net/questions/116123 | 27 | I asked this (with background) here
<https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision>
but did not really get any answers. See that post for the background.
Let $D$ be some open set in the plane, say. Not really important where the set $D... | https://mathoverflow.net/users/6494 | how to find/define eigenvectors as a continuous function of matrix? | The example given by Anthony Quas reveals a phenomenon discussed in Kato's book *Perturbation Theory for Linear Differential Operators*. The point is the following:
* If the symmetric matrix depends analytically upon **one** parameter, then you can follow analytically its eigenvalues and its eigenvectors. Notice that... | 34 | https://mathoverflow.net/users/8799 | 116153 | 65,684 |
https://mathoverflow.net/questions/116138 | 2 | How can I recover a Sidon set $A\subseteq \mathbb{Z}/n\mathbb{Z}$ from the set $A-A\subseteq \mathbb{Z}/n\mathbb{Z}$?
Is it even unique? (up to translation and reflection)
($A-A$ stands for the set of all differences of the form $a\_1-a\_2$ with both $a\_1$ and $a\_2$ in $A$ and a Sidon set is a set $A$ such that a... | https://mathoverflow.net/users/4878 | Recovering Sidon sets from difference sets | If $A$ is a Sidon set in $\mathbb{Z}/n\mathbb{Z}$ and $|A|=m$ then $A-A$ has $m^2+m+1$ members including $0.$ For $\lambda$ co-prime to $n$, $B=\lambda A$ is also a Sidon set and $B-B=\lambda(A-A).$ For ease I'll assume that $n$ is prime. We do have $B-B=A-A$ in the case that $A-A=\mathbb{Z}/n\mathbb{Z}.$ As noted, the... | 3 | https://mathoverflow.net/users/8008 | 116166 | 65,690 |
https://mathoverflow.net/questions/116178 | 2 | I have been reading through Wise's lecture notes on cubical complexes, which summarises the proof of the virtual Haken conjecture and the proof that all one-relator groups with torsion are residually finite.
My understanding of this stuff seems to have hit a wall.
Specifically, my understanding of (special) cubical... | https://mathoverflow.net/users/6503 | Cubical Complexes and Bass-Serre theory | I think your problem is solved if I tell you that the group is supposed to act freely (and specially, if you like) on a CAT(0) cube complex, or equivalently to be the fundamental group of a non-positively curved cube complex. In that sense, the theory only directly generalizes the fundamental groups of graphs, not grap... | 5 | https://mathoverflow.net/users/1463 | 116179 | 65,693 |
https://mathoverflow.net/questions/116155 | 13 | Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, and this question arose as a special case.
The question has a specific part, and a less specific part. Hopefully these... | https://mathoverflow.net/users/8133 | How should one look at the set of compatible ring structures on a given group? | Being a logician, I would look at this from the model theoretic point of view and think about [spaces of types](http://en.wikipedia.org/wiki/Type_%28model_theory%29), which are all rather nice [Stone spaces](http://en.wikipedia.org/wiki/Stone_space).
Here is the setup for your case for $0$-types. Let $\mathcal{L}\_G$... | 4 | https://mathoverflow.net/users/2000 | 116180 | 65,694 |
https://mathoverflow.net/questions/115088 | 8 | This is crossposted at stack exchange as <https://math.stackexchange.com/questions/248391/dirac-operators-on-s1>.
I am trying to understand the Dirac operators associated to the 2 spinor bundles on $S^1.$ I have been getting very confused about why one bundle has nontrivial harmonic spinors and the other doesn't.(Har... | https://mathoverflow.net/users/22781 | What are the Dirac operators on $S^1$? | This one is tricky and extremely confusing. The bundle of spinors is a complex line bundle so it is trivializable. You detect the spin structure only if you look at the Dirac operator. For one spin structure the Dirac operator has a kernel, for the other, it does not.
For a more detailed discussion on the pathologic... | 10 | https://mathoverflow.net/users/20302 | 116183 | 65,695 |
https://mathoverflow.net/questions/116162 | 1 | I suppose I am the first one who asked about Weil representation here.
In studying Weil representation, I fell into a slough and so determined to ask you for shedding a light. I think your responses would be a gleam of hope for many others struggling exactly at the same part like me.
Let me recall some basic notati... | https://mathoverflow.net/users/29877 | On the Weil representation of unitary groups. | At least as a place-holder answer: the issues may be more about just the geometric algebra rather than the Segal-Shale-Weil/oscillator repn. We can multiply hermitian and skew-hermitian forms by a non-zero element flipped in sign by Galois, making hermitian into skew and vice-versa, so there is no distinction between t... | 1 | https://mathoverflow.net/users/15629 | 116186 | 65,697 |
https://mathoverflow.net/questions/116200 | 13 | Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are *antipodal* if the origin lies in the segment that joins them.
**Question 1.** Assuming that $n$ is odd, does there always ... | https://mathoverflow.net/users/21123 | A problem on convex geometry | I will try to answer your question 2. The answer is negative. There must be at least two antipodal points with parallel tangent planes. Here is a sketch of a proof. The idea of that proof is that points you are looking for have variational nature. Proof works in any dimension.
Denote the boundary of your body by $\Ga... | 12 | https://mathoverflow.net/users/2823 | 116208 | 65,708 |
https://mathoverflow.net/questions/115949 | 27 | Recenly I came across [Peter Roquette](http://www.rzuser.uni-heidelberg.de/~ci3/)'s article [*On the history of Artin's $L$-functions and conductors*](http://www.rzuser.uni-hd.de/~ci3/lfunktio.pdf) (23 July 2003) in which he talks about some letters from Emil Artin and Emmy Noether to Helmut Hasse in the early 1930s.
... | https://mathoverflow.net/users/2821 | What happened to Emmy Noether's *Zukunftsphantasie* ? | I am not sure if I am interpreting the question correctly: is it whether it is possible to assign, naturally, an ideal $I\_\chi\subset O\_L$ to each $\chi$ in such a way that $N\_{L|K} I\_\chi=f(\chi,L|K)^{\chi(1)}$? Then the answer is No, even without the word "naturally", even for cyclotomic fields and even locally:
... | 12 | https://mathoverflow.net/users/3132 | 116210 | 65,709 |
https://mathoverflow.net/questions/116209 | 11 | How many balls have to be thrown uniformly at random into $m$ bins, such that with high probability $n\_1, n\_2, \dots, n\_m$ are distinct numbers, where $n\_i$ is the number of balls in bin $i$ ?
Is there anything known about this problem? A trivial lower bound is $m(m-1)/2$, as we need $m$ distinct values.
| https://mathoverflow.net/users/7368 | Balls and bins variation | This happens whenever $n\gg m^5$. To see this, notice that the expected number of balls in each bin is $n/m$ and the variance is also on the order of $n/m$. The distribution "tends" to N(n/m,n/m) (in the sense of CLT). We also have a local CLT, meaning that the for values $k\in\{n/m-\sqrt{n/m},\ldots,n/m+\sqrt{n/m}\}$ ... | 17 | https://mathoverflow.net/users/1061 | 116217 | 65,713 |
https://mathoverflow.net/questions/116215 | 15 | I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a simplicial (affine) scheme. To make the question reasonably self-contained and to give a sense of my background and curr... | https://mathoverflow.net/users/78 | Is there an algebraic "derived mapping space" construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces? | There is a standard answer to your question, unless I'm missing something, that I learned from the great [survey by Toen](http://arxiv.org/abs/math/0604504) and satisfies everything you want. However to satisfy your requirements - ie to get the correct Hochschild complex - you must (explicitly or implicitly) embed simp... | 12 | https://mathoverflow.net/users/582 | 116218 | 65,714 |
https://mathoverflow.net/questions/116171 | 3 | I read on page 4 [here](http://lipn.univ-paris13.fr/~toumazet/biblio/ARTICLES/NTB.pdf) that the Kostka coefficients $K\_{\lambda,\mu}$ are specializations of the Littlewood-Richardson coefficients $c^\tau\_{\sigma,\lambda}$ by specializing $\sigma,\tau$ depending on $\mu$ in a simple manner (certain sums of parts of $\... | https://mathoverflow.net/users/1056 | Skew Kostka coefficients from Littlewood-Richardson Coefficients | I prefer to write $K\_{\lambda/\nu,\mu}$ for $K^\nu\_{\lambda,\mu}$.
Using standard symmetric function notation, we have
$$ K\_{\lambda/\nu,\mu}=\langle s\_{\lambda/\nu},h\_\mu\rangle =
\langle s\_\lambda,s\_\nu h\_\mu\rangle. $$
Let $\rho/\sigma$ be a skew shape which is a disjoint union of shapes
$\nu, (\mu\_1), (\... | 7 | https://mathoverflow.net/users/2807 | 116224 | 65,717 |
https://mathoverflow.net/questions/39531 | 4 | Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie groupoid $G$ with $M$ as its space of objects, and here etale means that the source map is a local diffeomorphism). At least... | https://mathoverflow.net/users/4528 | Intrinsic Characterization of when an orbifold (or more general stack) is effective? | I answer this in <http://arxiv.org/abs/1212.2282>. An etale stack $\mathscr{X}$ is effective if and only if the substack assigning each manifold $M$ the groupoid of local diffeomorphisms $$M \to \mathscr{X}$$ is actually a sheaf, i.e. if and only if this groupoid is (equivalent to) a set.
| 2 | https://mathoverflow.net/users/4528 | 116240 | 65,724 |
https://mathoverflow.net/questions/116229 | 3 | In another posting I wrote about a trigonometric relation I had derived, but that ended up not being the main point of the posting:
[Strange pattern in rounding errors?](https://mathoverflow.net/questions/116214/strange-pattern-in-rounding-errors)
So as long as we're here, let's make it the main point of *this* pos... | https://mathoverflow.net/users/6316 | A "known" tangent half-angle formula? | Using the tangent double-angle formula $\tan\gamma=\frac{2\tan\tfrac{\gamma}{2}}{1-\tan^2\tfrac{\gamma}{2}}$ we get
$$\begin{align}
\tan\gamma & = \frac{2\tan\tfrac{\beta}{2}\tan\tfrac{\alpha}{2}}{1-\tan^2\tfrac{\beta}{2}\tan^2\tfrac{\alpha}{2}} \\[10pt]
& = \frac{2\sin\tfrac{\beta}{2}\sin\tfrac{\alpha}{2}\cos\tfrac{\b... | 2 | https://mathoverflow.net/users/20186 | 116242 | 65,726 |
https://mathoverflow.net/questions/116206 | 15 | The question is stated in the title, but I would like to add some motivation.
I've been teaching a course on complex tori and abelian varieties this semester and I would like to end it by showing some significant application of abelian varieties in algebraic geometry. I've come across a very beautiful recent proof by... | https://mathoverflow.net/users/10610 | How does one prove that the complete intersection of a quadric and a cubic of $\mathbb P^5$ is unirational? | You can look at the short paper by Conte, Marchisio end Murre [*On the k-unirationality of the cubic complex*](http://cab.unime.it/mus/429/) (2007).
It contains a proof of the unirationality of $V\_6$ over a field $k$ of any characteristic $\neq 2,3$, under the assumption that $V\_6$ has a $k$-rational point $p$ and... | 9 | https://mathoverflow.net/users/7460 | 116257 | 65,732 |
https://mathoverflow.net/questions/109260 | 5 | Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e$ be a nilpotent element in it. In the theory of finite W-algebras one often encounters the following two conditions:
1) $e$ is principal in some Levi subalgebra $\mathfrak l$ of $\mathfrak g$.
2) There exists a good even grading on $\mathfrak g$ ... | https://mathoverflow.net/users/3891 | Good even grading and principal Levi type | If $e$ is principal in a proper Levi subalgebra whose Dynkin diagram involves a component of type
$A\_k$ with $k$ odd, then e is not even. This is very easy to see by writing down an explicit $sl\_2$-triple containing $e$. Occasionally, one can find another good grading for such $e$ (if it is Richardson) but this is qu... | 4 | https://mathoverflow.net/users/24386 | 116259 | 65,734 |
https://mathoverflow.net/questions/116262 | 5 | A **real tree** is a metric space $(M,d)$ satisfying the following two conditions:
(1) for every $x,y\in M$, there is an unique isometry $\phi$ from the closed interval $[0,d(x,y)]$ onto $M$ such that $\phi(0)=x$ and $\phi(d(x,y))=y$; and
(2) any one-to-one continuous mapping $f:[0,1]\rightarrow M$ has the same ra... | https://mathoverflow.net/users/27566 | On multi-dimensional real trees | See the paper "Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings" MR1608566 by Kleiner and Leeb for a theory of $\mathbb{R}$-buildings that generalizes the theory of $\mathbb{R}$-trees. They use this theory for classifying, up to quasi-isometry, all symmetric spaces of noncompact type whose deRh... | 4 | https://mathoverflow.net/users/20787 | 116274 | 65,741 |
https://mathoverflow.net/questions/116269 | 17 | Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$.
Let $\Phi(w,z)$ be a polynomial in two variables, that is not identically zero. Assume that
$\Phi(0,0) =0$. Suppose $\textbf{formally}$ we have the identity
$$ \Phi(P(z),z) ... | https://mathoverflow.net/users/4463 | If a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence? | The equation $\Phi(w,z)=0$ can be solved using Puiseux series. If $\frac{\partial{\Phi}}{\partial{w}}\not\equiv 0$ then there exist finitely many formal series $f(z)=\sum\_{n\geq0}a\_nz^{n/p}$ such that formally $\Phi(w,z)=0$. All these series
are convergent. So the answer to your question is positive.
For the proof ... | 10 | https://mathoverflow.net/users/24309 | 116278 | 65,743 |
https://mathoverflow.net/questions/116283 | 1 | Hello
I'm sorry if this question is trivial but I haven't been able to find an answer. I'm trying to show that a sequence of distributions on $\mathbb{R}^n$ converges to the normal distribution by showing that the moments of the distributions converge to those of the normal distribution. Is this sufficient under approp... | https://mathoverflow.net/users/29907 | When does the limit of moments of multivariate distributions determine the limit distribution? | Yes, without any additional assumptions — the relevant technical conditions are satisfied because your limit distribution is normal. For sufficiency in the univariate case, see any probability textbook that covers the method of moments, for example section 30 of Billingsley's *Probability and Measure*. The multivariate... | 1 | https://mathoverflow.net/users/1044 | 116285 | 65,747 |
https://mathoverflow.net/questions/116268 | 0 | Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$ and $h$ are holomorphic functions.
Let us also assume that $Y$ is non empty. In particular
this would avoid somethi... | https://mathoverflow.net/users/4463 | Question regarding closure of sets defined by the vanishing of holomorphic functions | This is not true. Take $n=2$, $f(x,y)=x$, $ g(x,y)=y(y-1),\; h(x,y)=x-y$.
| 1 | https://mathoverflow.net/users/25510 | 116290 | 65,750 |
https://mathoverflow.net/questions/116292 | 8 | I'm relatively green in the differential geometry area, so my apologies if what I'm asking is ill-posed and/or not research-level.
I have a situation where I know the shortest path between any two points in the plane. Is there a way to reconstruct a corresponding 2D-manifold such that the shortest path between points... | https://mathoverflow.net/users/3400 | From Shortest Paths to Manifold Structure | **NB:** I'm assuming from the way you worded the question that your '2D manifold' is supposed to be a surface in $xyz$-space and that the 'projection' is the projection to the $xy$-plane. You may have had a more general situation in mind, such as a surface in $\mathbb{R}^n$ for $n>2$ or a more general projection than t... | 18 | https://mathoverflow.net/users/13972 | 116293 | 65,751 |
https://mathoverflow.net/questions/116256 | 4 | Two graphs are isospectral if the combinatorial Laplacian on them has the same spectrum, equivalently, the adjacency matrix has the same the set of eigenvalues (including multiplicities). Two graphs have the same matroid if they are 2-isomorphic, that means there exists a bijection between their edge sets that preserve... | https://mathoverflow.net/users/29902 | are there pairs of combinatorial graphs that are both isospectral and have the same matroid? | Choose a graph $X$ with vertices $u$ and $v$ such that $X\backslash u$ and $X\backslash v$ are cospectral. (In this case I say that $u$ and $v$ are cospectral vertices.) Assume that
there is no automorphism of $X$ that swaps $u$ and $v$. Now form the graph $Y$ from two
copies of $X$ by identifying vertex $u$ in the fir... | 3 | https://mathoverflow.net/users/1266 | 116296 | 65,752 |
https://mathoverflow.net/questions/116139 | 7 | Let $G$ be a classical group of dimension $n$ over $GF(q)$ where $q=p^f$ is a prime power, and $P$ be a Sylow $p$-subgroup of $G$. What is the maximal order of elements, i.e. the exponent, of $P$?
For $G=GL(n,q)$, it can be easily seen that the exponent of $P$ is the least power of $p$ greater than or equal to $n$. ... | https://mathoverflow.net/users/26700 | Exponent of Sylow $p$-subgroup of classical groups over a field of characteristic $p$ | The question is reasonable (even in the full generality of finite groups of Lie type in the defining characteristic $p$). However, the answer requires case-by-case study, as in related questions about classifying unipotent classes and centralizers of unipotent elements. Most of the information needed starts out in the ... | 7 | https://mathoverflow.net/users/4231 | 116298 | 65,754 |
https://mathoverflow.net/questions/116273 | 7 | Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n\_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y\_{\mathbf{Q}}$, defined by the vanishing of some homogeneous polynomials
$$
F\_1, \ldots, F\_k \in \mathbf{Z}[X\_0,\ldots,X\_n].
$$
How does one determine the set $S$ consi... | https://mathoverflow.net/users/17907 | How do you compute the primes of bad reduction? | Assume for simplicity that $Y$ has pure relative dimension $d$. By considering the standard affine cover of $\mathbf{P}^n\_{\mathbf{Z}}$, one easily reduces to the case where $Y=V(F\_1,\ldots,F\_k)$ is a closed subscheme of $\mathbf{A}^n\_{\mathbf{Z}}$. Then the special fiber $Y\_p = V(F\_{1,p},\ldots,F\_{k,p}) \subset... | 4 | https://mathoverflow.net/users/6506 | 116303 | 65,757 |
https://mathoverflow.net/questions/116302 | 2 | Let $M$ and $N$ be smooth manifolds and let $S$ be a submanifold of $N$ ($\dim S < \dim N$). Let $\mathfrak S$ be a foliation of $S$. We say that a map between $M$ and $N$ is transverse to $\mathfrak S$ if it is transverse to every leaf of $\mathfrak S$.
Now, suppose $f : M \rightarrow N$ is a smooth map transverse ... | https://mathoverflow.net/users/29330 | Smooth maps transverse to a foliation | I originally misunderstood your question. Here is a fairly simple example.
Let $N = \mathbb R^2$, $M=S^2$ and $S$ be the unit circle in $\mathbb R^2$ considered as foliated by its points -- the leaves are $0$-dimensional.
The map $M \to N$ will be a linear projection map. Let's project $S^2$ onto $\mathbb R^2$ in... | 1 | https://mathoverflow.net/users/1465 | 116307 | 65,759 |
https://mathoverflow.net/questions/116243 | 21 | I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is simple: let $G$ be a reductive (or even semisimple) algebraic group over $\mathbb Q$. Is it true that the adelic group... | https://mathoverflow.net/users/26116 | Is a reductive adelic group a Type I group? | I believe the answer is yes. Let's begin by recalling that if one wants to show that a locally compact group $G$ is of type I, it suffices to show that $G$ contains a "large" compact subgroup $K$, in the sense that for every $\pi \in \hat{G}$ and $\sigma \in \hat{K}$, the multiplicity of $\sigma$ in $\pi|\_K$ is finite... | 26 | https://mathoverflow.net/users/430 | 116323 | 65,769 |
https://mathoverflow.net/questions/116326 | 4 | Ryll-Nardzewski theorems states that if $T$ is a countable complete theory, then $T$ is $\aleph\_0$-categorical if and only if for every $n<\omega$ there are only finitely many formulas $\varphi(x\_1,\ldots,x\_n)$ up to equivalence relative to $T$.
$T$ is a countable theory if it can be built in a countable language.... | https://mathoverflow.net/users/29916 | Why Ryll-Nardzewski theorem fails for uncountable theories? | Let $T$ be the complete theory of $\mathbb N$, with a binary predicate $<$ for the standard ordering, unary predicates for all subsets of $\mathbb N$, and constants for all the elements of $\mathbb N$. This clearly has uncountably many inequivalent unary formulas. I claim that its standard model is, up to isomorphism, ... | 10 | https://mathoverflow.net/users/6794 | 116327 | 65,772 |
https://mathoverflow.net/questions/116328 | 11 | I am interested in the set $A$ of all positive integer numbers such that when factored into primes, the sum of the exponents is odd (I think of $A$ as the multiplicative odd numbers).
I want to know if it has positive upper density, more precisely
$$\bar d(A):=\limsup\_{n\to\infty}\frac{|A\cap[1,n]|}n$$
I think I rea... | https://mathoverflow.net/users/18698 | Density of the "multiplicative odd numbers" | Gerry has the right idea here: you are asking about the limiting behaviour of the sum
$$\frac{1}{x} \sum\_{n \leq x}{\frac{1 - (-1)^{\Omega(n)}}{2}}.$$
The arithmetic function $\lambda(n) = (-1)^{\Omega(n)}$ is known as Liouville's function. It is well-known (and equivalent to the prime number theorem!) that the summat... | 18 | https://mathoverflow.net/users/3803 | 116330 | 65,774 |
https://mathoverflow.net/questions/116312 | 18 | I would like to know the state of the art concerning the following two questions.
1) Does there exist a smooth 4-dimensional h-cobordism (so between closed 3-manifolds) with non-vanishing Whitehead torsion ?
2) Does there exist a smooth 4-dimensional s-cobordism (that is, with vanishing Whitehead torsion) which is ... | https://mathoverflow.net/users/29911 | 4-dimensional h-cobordisms | I think both questions are open. The somewhat sad state of affairs is that there are nontrivial TOP 4d s-cobordisms that are either nonsmoothable or not known to be smoothable, and there are smooth 4d s-cobordisms
that may well be products. No h-cobordisms with nontrivial torsion seems to be known.
It seems the state... | 12 | https://mathoverflow.net/users/1573 | 116335 | 65,777 |
https://mathoverflow.net/questions/116336 | 45 | At MIT all departments have numbers, and math is 18. Last year MIT
math majors produced a tee shirt that said ${i\choose 18}$ ("I choose
18") on the front, and on the back
$$ \frac{34376687+1499084559i}{14485008384}. $$
With the more natural denominator $18!$ this is
$$ \frac{15194495654000+662595375078000i}{18!}. $$... | https://mathoverflow.net/users/2807 | Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$ | **Asymptotics:** Lets look at the quantity
$$S(n)=(-1)^{n}(n+1)\binom{i}{n+1}=i\prod\_{k=1}^{n+1}\left(1-\frac{i}{k}\right).$$ It's just your binomial coefficient above with the $(-1)^{n+1}$ factored in, and an extra $n+1$ so it factors nicely as a product.
>
> **Claim:** We have that
>
>
> $$S(n)=\sqrt{\frac... | 27 | https://mathoverflow.net/users/12176 | 116346 | 65,780 |
https://mathoverflow.net/questions/53016 | 0 | Suppose for two given functions $f\_1,f\_2 \colon \mathbb{R}^2 \to \mathbb{R}$ there exist unique solutions $y\_1$ and $y\_2$ with the intersection of their intervals of existence $[0,\epsilon)$ to the integral equations $$y\_k(x)=\int\_{0}^{x} f\_k(t,y\_k(t)) dt$$. Moreover, suppose that $f\_1(x,y)\leq f\_2(x,y)$ (or ... | https://mathoverflow.net/users/1142 | Does $f_1(x,y)<f_2(x,y)$ imply $y_1<y_2$ for solutions to the integral equation $y_k'=f_k(x,y_k)$? | In general the answer seems to be no: Let
$$
f\_1(t,y)=\begin{cases} -1, &t\leq 0,\quad y\in \mathbb R\\\
1,& t>0,\quad y>t/2,\\\
-1,& t>0,\quad -t/2 \leq y \leq t/2,\\\
-e^{-n^2y},& t\in [1/n,1/(n-1)),\quad y< -t/2,\quad n\geq 1,
\end{cases}
$$
and
let $f\_2(t,y)=-f\_1(t,-y)$ so that $f\_1(t,y)< f\_2(t,y)$ when ... | 1 | https://mathoverflow.net/users/18410 | 116354 | 65,785 |
https://mathoverflow.net/questions/116344 | 3 | The following seems to be true: if $|W\_q| := \sum {q^{l(w)}}$, where the sum is taken over the elements $w$, then $|W\_q| = \prod {(1 + q +...+ q^{e\_i})}$, where the product is taken over the exponents $e\_i$.
In other words, if $V$ is the root space for $W$ and the polynomials $f\_k$ are the basis of $S[V]$ sur $S^W... | https://mathoverflow.net/users/29813 | For a Weyl group, what is the connection between its exponents and lengths of its elements? | I would leave this as a comment but I don't appear to have enough reputation points for that. Just to add to Philippe's answer that you will also find this as Theorem 10.2.3 in Carter's "Simple Groups of Lie Type", (it appears even earlier than this in Steinberg's Lecture Notes on Chevalley Groups, see Theorem 26 - pg.... | 5 | https://mathoverflow.net/users/22846 | 116355 | 65,786 |
https://mathoverflow.net/questions/116345 | 6 | Hi everyone.
Let $S$ be a closed surface with genus at least 3, $\alpha, \beta$ be the two vertices of
curve complex of $S$ such that $d\_{\mathcal {C}(S)}(\alpha, \beta)\geq 3$.
My question is
Is there a non-trivial finite ordered element $f$ of $MCG(S)$ such that $f(\alpha)=\alpha$ and $f(\beta)=\beta$ in $... | https://mathoverflow.net/users/18496 | The action of torsion of $MCG(S)$ on curve complex | Here is a way to find lots of examples. Suppose that $\Sigma$ is a surface and suppose that $f$ is a periodic mapping class. Let $S$ be the quotient orbifold $\Sigma/f$. Then taking full preimages gives a quasi-isometric embedding of the curve complex of $S$ into the curve complex of $\Sigma$. See
arXiv:1104.3492 an... | 6 | https://mathoverflow.net/users/1650 | 116357 | 65,787 |
https://mathoverflow.net/questions/116275 | 5 | The $n$-th Bell number $B\_n$ represents the number of distinct partitions of a set with $n$ distinguished elements.
It can be expressed as the infinite sum $B\_n = (1/e)\sum\_{k=1}^{\infty} (k^n/k!)$, which is also the $n$-th moment of a Poisson distribution with mean $1$.
The first few values are known precisely; the... | https://mathoverflow.net/users/7252 | Simple lower bounds for Bell numbers (number of set partitions)? | On further reflection, it seems the answer is no.
By considering the most significant terms in the asymptotic analysis of de Bruijn, and arguing that they dominate the other terms for large enough $n$, it seems possible to show that for every $\epsilon > 0$, there is some threshold $n\_0 = n\_0(\epsilon)$ such that
$... | 2 | https://mathoverflow.net/users/7252 | 116371 | 65,791 |
https://mathoverflow.net/questions/116366 | 5 | Hello.
I am working on investigation of family of dynamical systems on the torus
$$\dot{x}=\cos(x)+b\cos(t)+a$$
$$\dot{t}=1$$
and it's Poincare map $$P:(x,0) \rightarrow (P(x),2\pi=0)$$
I need to find Arnold tongues of map $P$. I tried simple calculation of solution using Runge-Kutta formulas, then iterating and checki... | https://mathoverflow.net/users/29934 | Numerical calculation of Arnold tongue | There is a good way to compute rotation number of a circle homeomorphism (this was the way Poincaré thinked of it): you calculate the rotation number buy its continued fraction in a direct way.
You start from a point $x$ and $f(x)$: this gives you a decomposition of the circle into points that are on the right side o... | 8 | https://mathoverflow.net/users/47274 | 116374 | 65,793 |
https://mathoverflow.net/questions/116370 | 2 | Given a family F of subsets of [n]={1,2,3,...,n}.
If the cardinality of the intersection of two arbitrary elements of F is not greate than a predefined value S-MIN.
Then what is the sum of cardinalities of all elements of F at most?
| https://mathoverflow.net/users/29936 | what is the sum of cardinalities of all elements of F at most? | If $A\in F$ has more than $s+1$ elements we may remove $A$ from $F$ and insert instead all the sets $C\subset A$ of cardinality at most $s+1$. Some may already be in $F$, but not those of cardinality $s+1$, forbidden in $F$ because of their intersection with $A$ (themselves), too large. So this increases $\sum \_ {A\in... | 2 | https://mathoverflow.net/users/6101 | 116376 | 65,795 |
https://mathoverflow.net/questions/115004 | 7 | Completely unaware of the Bohr topology, I recently [asked](https://mathoverflow.net/questions/114816/hausdorff-group-topologies-on-finitely-generated-groups) whether or not there was a Hausdorff group topology on the integers $\mathbb{Z}$ which made the group fail to be first countable. For me, this topological group ... | https://mathoverflow.net/users/5801 | The integers as a sequential but non-first countable topological group | The answer is no.
It is proved in *Topologies on Abelian Groups* (E.G. Zelenyuk and I.V. Protasov, Math. USSR Izvestiya, 1991), that on every infinite abelian group there exists a sequential Hausdorff group topology which is not first-countable.
| 8 | https://mathoverflow.net/users/17836 | 116385 | 65,799 |
https://mathoverflow.net/questions/116383 | 10 | EDIT The question was based on an error, as it turns out. In fact my example is a category (and therefore a groupoid), by Eric Wofsey's argument. I can't remember why I thought it wasn't, and I feel a little silly, but I am very glad that it is.
In the nerve of a small category an $n$-simplex is determined by $n+1$ ... | https://mathoverflow.net/users/6666 | This is not a category. What is it? | It's called a groupoid. Given an object $A$, call the degenerate edge from $A$ to itself the identity map at $A$. Given an edge $f:A\to B$, let $f^{-1}:B\to A$ denote the unique edge that fills in a 2-simplex whose other two edges are $f$ and the identity. Given $f:A\to B$ and $g:B\to C$, define $gf$ to be the unique e... | 19 | https://mathoverflow.net/users/75 | 116391 | 65,800 |
https://mathoverflow.net/questions/116395 | 7 | Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector space over $E$ and $q$ is a hermitian form on $V$. By a decomposition theorem, $V$ decomposes as a sum of hyperbolic pla... | https://mathoverflow.net/users/13027 | Classification of quasi-split unitary groups | There is a text by Scharlau about "Hermitian...". Also the older book by O'Meara.
The point is that, first, over non-archimedean local fields a *quadratic* form in five or more variables has an isotropic vector. In case the residue characteristic is not two, this has a reasonably elementary direct proof. Then note th... | 8 | https://mathoverflow.net/users/15629 | 116406 | 65,809 |
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