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https://mathoverflow.net/questions/95970 | 3 | Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let
$W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional
simplices (i.e. bases of the affine matroid $\mathcal{M}(S)$ of $S$) with vertices in $S$. In particular
$\dim W(S) =$ number of simplices= number of... | https://mathoverflow.net/users/11100 | from affine matroid to measures | This conjecture is proven in theorem 7.4 of ["Incidence matrices, geometrical bases, combinatorial prebases and matroids"](http://www.sciencedirect.com/science/article/pii/S0012365X97001076) by T.V. Alekseyevskaya and I.M. Gelfand ($n=2$) and theorem 4.5 in ["Bases in Systems of Simplices and Chambers"](http://arxiv.or... | 3 | https://mathoverflow.net/users/2384 | 95976 | 56,153 |
https://mathoverflow.net/questions/95974 | 21 | It is a well-known and deep${}^\ast$ theorem that if a group $G$ has cohomological dimension one then it must be free. This was proved in the late 60's by Stallings (for finitely generated groups) and Swan.
The proofs in the original articles are well-written and informative, bringing together a lot of ideas from top... | https://mathoverflow.net/users/8103 | Proofs of the Stallings-Swan theorem | The heart of the matter is the Stallings' "ends of groups" theorem: A finitely-generated group with infinitely many ends splits as graph of groups with finite edge groups. In addition, one also has to show that the decomposition process terminates for your group (this property is called *accessibility*). Neither one ha... | 34 | https://mathoverflow.net/users/21684 | 95983 | 56,157 |
https://mathoverflow.net/questions/95909 | 0 | I was working with symplectic submanifolds when I posed the following question:
Suppose I have a Hamiltonian system with the phase space $\mathcal{M}$, a symplectic manifold with the standard symplectic form. Now assume that the Hamiltonian system has two first integrals $C\_1,C\_2$. Define the restricted phase space... | https://mathoverflow.net/users/23433 | Symplectic submanifolds and first integrals | I am posting here the answer that I gave to the same question when it was posted yesterday on [MSE](https://math.stackexchange.com/questions/140470/symplectic-submanifolds-and-first-integrals).
Let $f\_1$ and $f\_2$ be independent functions on a symplectic manifold $(M,\omega).$
Let us denote by $\Sigma$ the subma... | 0 | https://mathoverflow.net/users/12617 | 95987 | 56,160 |
https://mathoverflow.net/questions/95969 | 2 | Jones Lemma is One scale about recognizing that a topological space is not normal.
This lemma tells us, if The topological space $X$ has a dense subset $D$ and a closed discrete subset $S$, with the property that $2^{|D|} \le|S|$, it couldn't be a normal space. But I think there is no apparent counterexample about the ... | https://mathoverflow.net/users/23317 | Counterexample about Jones lemma with special weak condition. | Yes, take, for example the Sorgenfrey plane $P$. A standard example of a non-normal space. It is separable and its anti-diagonal $\lbrace (x,-x):x\in\mathbb{R}\rbrace$ is closed and discrete, so Jones' Lemma is applicable in this case. Take any Hausdorff compactification of $P$; the result is a separable normal space a... | 4 | https://mathoverflow.net/users/5903 | 95990 | 56,162 |
https://mathoverflow.net/questions/95961 | 1 | We know that Kahler curvature tensor can be decomposed into three items:scalar part, traceless Ricci part and Bochner curvature tensor. In page 77 of Besse's book, it appears two symbols: $B$ and $B\_0$. I don't know which one stands for exactly the Bochner curvature tensor. If someone is Bochner tensor curvature, then... | https://mathoverflow.net/users/19071 | about Kahler curvature tensor on page 77 of Besse's book "Einstein Manifolds" | Well, $B$, as defined in Besse on page 77, is not the Bochner curvature because it is not traceless when $m>1$ (cf. (2.64)). I believe that $B\_0$ is some version of the Bochner curvature tensor.
Incidentally, the decomposition can be understood a little better by realizing that the space of curvature tensors of Käh... | 4 | https://mathoverflow.net/users/13972 | 95991 | 56,163 |
https://mathoverflow.net/questions/95727 | 6 | Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, and $\mathcal{O}\_N$ be an Eichler order of level $N$. Is there an element $x\in \mathcal{O}\_N$ such that its reduced norm $\mathrm{Nrd}(x)=-1$?
There exists an element $x\in B$ such that $\mathrm{Nrd}(x)=-1$. This follows from the no... | https://mathoverflow.net/users/5738 | Existence of elements in an Eichler order whose norm is minus one | I can think of two ways to answer your question.
The first is to revisit what it is I think you want to conclude, which is that every right fractional O-ideal is (principal and) generated by an element with positive reduced norm. It is in this way that the question most naturally arises when considering double coset ... | 4 | https://mathoverflow.net/users/4433 | 95992 | 56,164 |
https://mathoverflow.net/questions/95993 | 8 | I am trying to understand the proof of Proposition 4 in
S. Ullom, *Integral normal bases in Galois extensions of local fields*, Nagoya Math. J. Volume 39 (1970), 141-148. The PDF is available here:
<http://projecteuclid.org/euclid.nmj/1118798052>
It appears that the following result is used, but I'm afraid that I don... | https://mathoverflow.net/users/7443 | Representation theory of a finite p-group over a field of characteristic p: dim of invariants =1 => dim of coinvariants = 1? | The point is that you can inject $W$ into $k[G]$ and because of the equality of dimension this is an iso. This lets you to conclude that the coinvariants are $1$-dimensional.
The injection goes as follows:
$k[G]$ is a projective $k[G]$-module for trivial reasons, so its $k$-linear dual is injective, but $k[G]^\*\c... | 9 | https://mathoverflow.net/users/23457 | 95998 | 56,168 |
https://mathoverflow.net/questions/95982 | 12 | The exceptional complex simple Lie algebra $F\_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in Humphreys's book on Lie algebras and representation theory. In fact, $F\_4$ can be defined as the Lie subalgebra of $\mathfra... | https://mathoverflow.net/users/394 | Type of 26-dimensional representation of different real forms of the complex simple Lie algebra $F_4$ | I think the best way to see the signature of these quadratic forms is by using the formula from "A Classification Theorem for Albert Algebras" by R. Parimala, R. Sridharan, and Maneesh L. Thakur, Trans. AMS 350 #3, March 1998.
All forms of $F\_4$ arise from Albert algebras. Over $R$, these are 27-dimensional algebras... | 12 | https://mathoverflow.net/users/3545 | 96008 | 56,171 |
https://mathoverflow.net/questions/96006 | 11 | Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates points, i.e. there is no non-trivial measurable partition of $X$ such that each function in $A$ is constant on (almost ... | https://mathoverflow.net/users/4312 | Stone-Weierstrass analogue for $L^p$ | Yes (I assume that the measure is finite). Here is a proof that uses the von Neumann bicommutant theorem (or rather [Kaplansky's density theorem](http://en.wikipedia.org/wiki/Kaplansky_density_theorem)).
See $A \subset L^\infty(X,\mu) \subset B(L^2(X,\mu))$ where $L^\infty$ acts on $L^2$ by pointwise multiplication. ... | 9 | https://mathoverflow.net/users/10265 | 96025 | 56,180 |
https://mathoverflow.net/questions/96023 | 0 | Let $X=\{0,1\}^{\mathbb{N}}$ and $\xi\_n$ be the partition of $X$ defined by the equivalence relation $x \sim\_n x' \Leftrightarrow (x\_{n}, x\_{n+1}, \ldots) = (x\_{n}', x\_{n+1}', \ldots)$. The sequence of partitions $(\xi\_n)$ is decreasing and we introduce the intersection partition $\theta= \cap \xi\_n$. I'm loook... | https://mathoverflow.net/users/21339 | intersection partition as an orbital partition | Consider the group $G\_n$ of all permutations of strings $\{0,1\}^{\{1,2,...,n\}}$ of length $n$. That group acts on the set of all strings $\{0,1\}^{\mathbb N}$ in the natural way (it changes the $n$-prefix only). Now clearly $G\_n\subset G\_{n+1}$ for every $n$. Let $G=\cup G\_n$. It is a (locally finite) group which... | 1 | https://mathoverflow.net/users/nan | 96026 | 56,181 |
https://mathoverflow.net/questions/95564 | 5 | We denote the rings of all real valued continuous functions on compeletely regular Hausdorff space $X$ by $C(X)$.Let $I$ be a ring ideal of $C(X)$. define $$Z[I]:=\lbrace Z(f):\;f\in I\rbrace$$ where $Z(f):=\lbrace x \in X:\;f(x)=0\rbrace$. We are interested in knowing about the relation about algebraic properties of t... | https://mathoverflow.net/users/23317 | Ideals of $C(X)$ with only finitely many number of zerosets | **Edit:** In the first version I had the additional assumption that each singleton in $X$ is a zero set. But as observed by AliReza Olfati the proof can be adapted to work in the general case. Therefore we have:
>
> $Z(I)$ is finite if and only if the following holds:
>
>
> 1. There is $A \subseteq X$ closed an... | 5 | https://mathoverflow.net/users/10194 | 96029 | 56,183 |
https://mathoverflow.net/questions/96027 | 0 | Hello,
If I'm not mistaken, globally speaking, Riemann's explicit formula establishes a duality between prime numbers and the non trivial zeroes of the Riemann zeta functions. The imaginary parts of these zeroes are widely believed, assuming RH, to be linearly independent over the rationals.
My question is: is a vio... | https://mathoverflow.net/users/13625 | Linear (in)dependence of $\Im(\rho_n)$ and fundamental theorem of arithmetic | The duality to which you refer does not mean that linear dependence of the zeros is equivalent to linear dependence of primes (or even their logarithms.)
You might look at Odlyzko and te Riele's disproof of the Mertens' conjecture, and the related literature - the truth of Mertens would have implied linear dependence... | 7 | https://mathoverflow.net/users/6756 | 96030 | 56,184 |
https://mathoverflow.net/questions/95805 | 3 | This is a question about the definition of a smooth function in Guillemin and Pollack's "Differential Topology". G&P define all manifolds as objects embedded in $\mathbb{R}^N$ for some $N$, which (as a zillion people have reminded me) is not how mathematicians usually think of manifolds - nevertheless, this is a questi... | https://mathoverflow.net/users/23404 | On The Definition of Smoothness in "Differential Topology" by Guillemin & Pollack | when you read a little further, page 8 or so, you will find that the derivative of a smooth map is a linear map defined only on vectors tangent to the manifold. it will turn out that all extensions of the smooth function act the same on vectors tangent to the manifold. In your example both extensions y and 0 act as the... | 7 | https://mathoverflow.net/users/9449 | 96035 | 56,186 |
https://mathoverflow.net/questions/96036 | 0 | Looking for a general solution to the following second-order PDE, where the unknown is a function $f(x\_1, x\_2)$ of two variables:
$
0=a^2f+a^2x\_1{\partial f\over \partial x\_1}+b^2x\_2{\partial f\over \partial x\_2}+\frac{1}{2}
\left(a^2{\partial^2 f\over \partial x\_1^2} + 2ab{\partial^2 f\over \partial x\_1\part... | https://mathoverflow.net/users/23470 | Solution to a second order PDE | If you take the Fourier transform, the equation reduces to a first order hyperbolic equation, which can then be solved by the method of characteristics.
| 4 | https://mathoverflow.net/users/12120 | 96039 | 56,189 |
https://mathoverflow.net/questions/96041 | 4 | Is there an analog for the Tate-Shafarevich group for hyperelliptic curves?
References to such an analog would be nice if one exists.
EDIT:
Referring to Noam Elkies' comment, are there any finiteness conjectures for such an analog?
| https://mathoverflow.net/users/22095 | Analog for Tate-Shafarevich group | There are Tate-Shafarevich groups for every number field $K$ and every smooth locally algebraic group scheme $G$ over $X \setminus S$ where $X$ is the spectrum of the ring of integers in $K$ and $S$ is a finite set of places containing all infinite places. In this case, the Tate-Shafarevich "groups" (actually they are ... | 6 | https://mathoverflow.net/users/10194 | 96047 | 56,193 |
https://mathoverflow.net/questions/96044 | 1 | Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f\_n \rightarrow f$ weakly in $X$, and $\|f\_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly convex Banach space, then we have $f\_n \rightarrow f$ strongly in $X$. Is this true for an reflexive Banach space?
| https://mathoverflow.net/users/23355 | Strong convergence in reflecxive Banach space | I take it "strongly" means "in norm"? No, this can fail for reflexive Banach spaces.
For a counterexample, let $X$ be the $l^\infty$ direct sum of ${\bf R}$ and $l^2$. (I'm doing this for real Banach spaces, but the same counterexample works in the complex case too.) Thus a typical element of $X$ looks like $(a,f)$ w... | 7 | https://mathoverflow.net/users/23141 | 96051 | 56,195 |
https://mathoverflow.net/questions/96050 | 3 | Fix a standard effective listing $(\phi\_e)\_{e\in\omega}$ of the partial computable functions from $\omega$ to $\omega$. Let $\mathcal{C}$ be a class of computable total functions $\omega\rightarrow \omega$. Say that a set $X\subseteq\omega$ is a *representative* for $\mathcal{C}$ if $e\in X\implies \phi\_e\in\mathcal... | https://mathoverflow.net/users/8133 | Diagonalization and classes of computable functions | The set of polytime computable functions does indeed have a computable representative. Note that for $f$ to be polytime computable means there exists $t$, and $n\_0$ such that for all $n\ge n\_0$ it takes only at most $n^t$ time steps to compute $f(n)$. Now dovetail over all indices $e\in\omega$ and all $t$ and $n\_0$;... | 3 | https://mathoverflow.net/users/4600 | 96052 | 56,196 |
https://mathoverflow.net/questions/96063 | 3 | [SOME PROPERTIES OF THE SERIES OF COMPOSED NUMBERS, p2](http://www.emis.de/journals/EM/expmath/volumes/11/11.2/Garunkstis301.pdf) gives the bounds
$$l(x)=\frac{x}{\log(x)-28/29}<\pi(x) < u(x)=\frac{x}{\log(x)-1.12} \qquad (1)$$
for $x \geq 3299$.
[TWO GENERALIZATIONS OF LANDAU’S INEQUALITY, p6](http://files.ele-m... | https://mathoverflow.net/users/12481 | Can these logarithmic inequalities show existence of a prime between (x-1)^2 and x^2 | Note that, in the second paper, the inequality $$x\pi(x)<2\pi(x^2)$$
is derived from the inequality
$$xy\pi(x)\pi(y)>4\pi^2(xy)$$
by setting $x=y$. The proof for theorem 3 seems correct to me, so the sign flip is definitely a typo :)
| 5 | https://mathoverflow.net/users/2384 | 96064 | 56,198 |
https://mathoverflow.net/questions/96060 | 0 | I am trying to argue that exterior measure has nice properties that Jordan outer measure doesn't have. One of them is finite additivity, but I can't find a simple way to show Jordan outer measure is not finitely additive on positively separated sets in $\mathbb{R^n}$? Can someone give me a simple proof or a counter exa... | https://mathoverflow.net/users/23475 | Is Jordan outer measure finitely additive on positively separated sets in $\mathbb{R^n}$? | It is finitely additive for separated sets. A sufficiently small cube can not
cut both sets.
| 2 | https://mathoverflow.net/users/7402 | 96065 | 56,199 |
https://mathoverflow.net/questions/96062 | 2 | I want to distinguish $S^2$-bundles over $S^2$ from $CP^2\sharp CP^2$.
As you know, a $S^2$-bundle over $S^2$ is $S^2\times S^2$ or $M= S^3\times\_{S^1}S^2$ where $M$ is diffeomorphic to a cohomogeneity one manifold, i.e, $M/G=[0,1]$, whose group diagram is $G=S^3 \supset K\_- = S^1, K\_+ = S^1 \supset H=\{ 1\}$.
... | https://mathoverflow.net/users/36572 | Difference between $S^2$-bundles over $S^2$ and $CP^2\sharp CP^2$ | **Summary about the manifolds $S^2\times S^2$, $S^2\tilde\times S^2$ and $\mathbb CP^2\#\mathbb CP^2$:**
1. There are only two $S^2$-bundles over $S^2$: the trivial bundle $S^2 \times S^2$ or the non-trivial bundle usually denoted $S^2\tilde\times S^2$, which is diffeomorphic to $\mathbb C P^2\#-\mathbb C P^2$, i.e.,... | 3 | https://mathoverflow.net/users/15743 | 96073 | 56,201 |
https://mathoverflow.net/questions/96068 | 1 | Let $X=\{0,1\}^{\mathbb{N}}$ and $\theta$ be the partition of $X$ induced by the equivalence relation $x \sim x'$ when $x$ and $x'$ differ only at a finite number of coordinates (see [this related question](https://mathoverflow.net/questions/96023/intersection-partition-as-an-orbital-partition)).
Given a Bernoulli me... | https://mathoverflow.net/users/21339 | ergodicity of the group of transformations preserving a partition | It's easy to see that the action is always ergodic since $\mathcal G$ contains the group of finite permutations on the indices, which acts ergodically. In fact, the group $\mathcal G$ (which equals $\mathcal H$ in the case $m = \mu^{\mathbb N}$ with $\mu(0) = 1/2$.) that you are describing is the full group of the ergo... | 3 | https://mathoverflow.net/users/6460 | 96075 | 56,203 |
https://mathoverflow.net/questions/96070 | 2 | Let $L\_D(\mathbb{R}^n)^n$ be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is
$$L\_D(\mathbb{R}^n)^n=\{Du\colon u\in H^1\_{loc}(\mathbb{R}^n), Du\in L^2(\mathbb{R}^n)^n\}.$$
For $n>2$ it can be shown that $C\_c(\mathbb{R}^n)^n\cap L\_D(\mathbb{R}^n)^... | https://mathoverflow.net/users/21962 | Is $C_c(\mathbb{R}^2,\mathbb{R}^2)$ dense in the irrotational square integrable functions? | It is true even for n=1. Consider the Fourier transform of functions u which have $Du\in L\_D^n$. For such functions we have
$$\int |k|^2\hat u(k)^2\,dk<\infty.$$
The norm associated with this integral corresponds to the norm in $L\_D^n$. In this norm, we can approximate $\hat u$ by a test function; first cut off near ... | 2 | https://mathoverflow.net/users/12120 | 96079 | 56,205 |
https://mathoverflow.net/questions/96078 | 31 | Products, are very elementary forms of categorical limits. My question is whether in the category of groups, semi-direct products are categorical limits.
As was pointed in:
<http://unapologetic.wordpress.com/2007/03/08/split-exact-sequences-and-semidirect-products/>
Bourbaki (General Topology, Prop. 27) gives a uni... | https://mathoverflow.net/users/5309 | Are semi-direct products categorical (co)limits? | This is a partial answer, summing up some of my comments.
The semi-direct product is not a limit, but rather it is a colimit. The reason is that the universal property cited above describes maps *on* the semi-direct product. In the special case that $\phi$ is the trivial action, the semi-direct product becomes the di... | 39 | https://mathoverflow.net/users/2841 | 96081 | 56,207 |
https://mathoverflow.net/questions/96042 | 6 | Is the eta invariant of spherical space form $\eta(S^3/\Gamma)$ always nonegative?
Can we calculate it with the information of $\Gamma\in SO(4)$ explicitly?
In fact, i need a reference for the calculation of eta invariant. Can some one give me some advice or download the following paper for me? Thank you!
(1) Hitc... | https://mathoverflow.net/users/22880 | Eta Invariant of Spherical Space Form | This is not a full answer, but for the lens spaces (i.e., $\Gamma$ cyclic), the eta invariant is computed in the [second paper](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2075856) of Atiyah, Patodi and Singer. In case you're outside the paywall, you can try Proposition 5.2 in [this eprint]... | 6 | https://mathoverflow.net/users/394 | 96082 | 56,208 |
https://mathoverflow.net/questions/96095 | 8 | I am studying Kunen's Set Theory (2011 edition) on my own. I got stuck at the excercise I.9.6 which is:
**Excercise I.9.6.** Derive the axiom of replacement from lemma I.9.5.
And the mentioned lemma is this:
**Lemma I.9.5.** For a relation R and a class A, if R is set-like on A, then R\* is set-like on A.
Here ... | https://mathoverflow.net/users/23484 | An exercise in Kunen. Getting Axiom of Replacement from set-like transitive closure. | Suppose that the lemma holds and that we are considering an instance of the replacement axiom, so we have a set $X$ and for some parameter $z$ and for every $x\in X$ there is a unique $y$ such that $\varphi(x,y,z)$. Fix any set $w$ not in $X$, and let $R$ be the class relation such that $R(x,w)$ for each $x\in X$, and ... | 15 | https://mathoverflow.net/users/1946 | 96098 | 56,214 |
https://mathoverflow.net/questions/96100 | 1 | Consider then a convex subset of probability measures $A$ that is closed relative to the set of all probability measures. I'm wondering if $\forall \mu \in$ closure($A$), does there exists a positive measure $\nu$ such that $\mu + \nu \in A$?
It is well known that $C^\\*\_0(\mathbb{R})$ (the continuous dual space of... | https://mathoverflow.net/users/23487 | Probabilities Measures | This is false. Here's a counterexample.
For $n \geq 2$, let $\mu\_n$ be the probability measure $\mu\_n = (1/n)\delta\_0 + (1/2 - 1/n)\delta\_1 + (1/2)\delta\_n$. The convex hull of $\{\mu\_n: n \geq 2\}$ is contained in the set $P$ of probability measures; let $A$ be its closure in $P$ for the relative weak\* topolo... | 3 | https://mathoverflow.net/users/23141 | 96105 | 56,217 |
https://mathoverflow.net/questions/96099 | 1 | Hello,
is it possible do somehow define regular function(in the way that some analogy to Cauchy integral formula would hold) over quaternions that function of form $$f(q)= \sum\_{n=0}^\inf a\_n q^n b\_n$$ $a,b,c,q \in \mathbb{H}$, would be regular?
With classic definition of regular function linear functions $(f(q)... | https://mathoverflow.net/users/21000 | regular quaternion functions of form f(q) =c + a*q*b + ... | Check out <http://www.zipcon.net/~swhite/docs/math/quaternions/analysis.html> and references therein.
| 1 | https://mathoverflow.net/users/11142 | 96114 | 56,221 |
https://mathoverflow.net/questions/96118 | 3 | I found that related to the Kähler cone there are many discussions on MathOverflow.
Recently I am interested in the very special manifold of the one-point blow up of $\mathbb{C}P^n$ and just want to see what the general results on Fano Kähler manifolds look like when it comes to this special manifold.
My question is... | https://mathoverflow.net/users/19071 | What does the Kähler cone of the one-point blow-up of $\mathbb{C}P^n$ look like? | The Kahler cone of any compact manifold is described by a theorem of [Demailly and Paun](http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/kahlercone.pdf). If $X$ is a compact Kahler manifold, then its Kahler cone is one of the connected components of the set
$$
\mathcal P =
\lbrace \alpha \in H^{1,1}(X,\mathbb... | 5 | https://mathoverflow.net/users/4054 | 96119 | 56,224 |
https://mathoverflow.net/questions/96137 | 18 | Background on the Adèles
------------------------
The Adèles $\mathbb{A}\_K$ of a number field or function field $K$ are defined as a *restricted product* of the complete local fields $K\_\nu$, where $\nu$ ranges over all places of $K$. The restricted product is usually defined as the subset of $\prod\_\nu K\_\nu$ gi... | https://mathoverflow.net/users/956 | Categorical description of the restricted product (Adeles) | I was puzzled by this question as well a while ago. Here is a suggestion, which worked for me:
1. The product and the co-product of categories are best defined by an universal mapping property.
2. The adeles $\mathbb{A}$ are the inductive limit of all $S$-adeles
$$\mathbb{A}(S) = \mathbb{R} \times \prod\limits\_{p ... | 8 | https://mathoverflow.net/users/10400 | 96138 | 56,230 |
https://mathoverflow.net/questions/95994 | 0 | Consider a vector space $V$ (with dimension $n+1$ and elements $v$) on which a (commutative and associative) "product" $\odot$ taking $V\odot V\rightarrow V$ is defined, and an $1$ element $v\_0$ exists: $v\odot v\_0=v$ for all $v\in V$. $\odot$ is now completely defined by choosing a basis $v\_m$ $(0\le m\le n)$ ($v\_... | https://mathoverflow.net/users/11504 | Vector "product" diagonalization | Assuming you want $(A+B) ⊙ V = A⊙V + B⊙V$ (if not things get a lot more complicated , you might as well ignore the vector structure having no way to link the addition and product) ,
the product , being bi-linear , can be described as a tensor (essentially n square matrices $M\_p$, n\*n each , such that if $V\_1⊙V\_2 =C... | 0 | https://mathoverflow.net/users/23222 | 96139 | 56,231 |
https://mathoverflow.net/questions/96125 | 6 | As you know, the Hopf conjecture is about the existence of positively curved metric on $S^2\times S^2$. Hsiang-Kleiner have shown that there exists no positively curved metric admitting $S^1$-action on $S^2\times S^2$.
My question is simple. If $(S\_1=S^2,g)$ and $(S\_2=S^2,h)$ are positively curved,
then for any ... | https://mathoverflow.net/users/36572 | Positively curved metrics on $S^2\times S^2$ | As already observed above, if you consider a warped product metric $g\_1 + f \cdot g\_2$ on $S^2\times S^2$ obtained from a positively curved metric on each factor, it **will not have positive curvature**. This can be seen in the following way. The formula for the sectional curvature of "mixed planes" (i.e., spanned by... | 8 | https://mathoverflow.net/users/15743 | 96145 | 56,235 |
https://mathoverflow.net/questions/95880 | 5 | Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma\_B$ be the compact core of the $B$-covering space of $S$ (well-defined up to isotopy in the $B$-covering space). Lifting hyperbolic structures gives a well-d... | https://mathoverflow.net/users/20787 | Does the fundamental group of a surface have rigid subgroups? | Regarding Question 2, you get lots of examples that are rigid for the lifting map $M(\Gamma) \to M(\Gamma\_B)$.
Let $B$ be finitely generated subgroup of $\Gamma$ (considered a fuchsian group) such that every nontrivial element of $B$ fills $S$ (there are lots of these). Proposition 5.1 of [Kent, Leininger, and Schl... | 2 | https://mathoverflow.net/users/1335 | 96159 | 56,240 |
https://mathoverflow.net/questions/96162 | 3 | The following series evaluation
$\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$
seems attractive to me, and has a proof related to the evaluation of $\zeta(2)$.
Does this identity (and/or it's many variants) occur in the literature? If so, in what context?
Also, does there exist a close... | https://mathoverflow.net/users/10909 | $\sum \frac{n^2-1}{(n^2+1)^2}=\frac{1}{2}(1-\frac{\pi^2}{\sinh(\pi)^2})$ | This is a special case of
$$\sum\_{n=-\infty}^\infty f(n)=-\sum\_{poles\ of\ f} Res(\pi\cot(\pi z)f(z)).$$
The formula is true for rational functions for which the series converges. Proofs can be found in complex analysis texts
| 9 | https://mathoverflow.net/users/12120 | 96167 | 56,244 |
https://mathoverflow.net/questions/96165 | 3 | Suppose I have two variables $X$ and $Y$ which have continuous p.d.f.s $f$ and $g$ on the positive real line. I know that the moments $\mathrm{E}[X^n] > \mathrm{E}[Y^n]$ for sufficiently large $n$ (and all moments exist). Can I conclude that there exists a $\xi$ s.t. $f(x) > g(x)$ for all $x > \xi$?
| https://mathoverflow.net/users/8537 | Comparing distributions with moments | Certainly not. Let g be any probability distribution function satisfying g(0)=0 and let
f(x)=g(x-1) if x>1 and f(x)=0 if x<1. Then
$$\int\_0^\infty f(x)x^n\,dx=\int\_0^\infty g(x)(x+1)^n\,dx>\int\_0^\infty x^ng(x)\,dx$$
for any n. But it is not necessarily true that g(x-1)>g(x) for large x.
| 4 | https://mathoverflow.net/users/12120 | 96168 | 56,245 |
https://mathoverflow.net/questions/95981 | 4 | A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.
Is there a more abstract characterisation of this property?
I am looking for something that looks like a diagram, a functor or anything vaguely related (... | https://mathoverflow.net/users/11084 | Abstract definition of properly discontinuous action | (This answer started out as a comment, but got too long, and perhaps there is something that may be gleaned that will help the OP)
The problem with this question for me is that you are assuming that such a thing *can* be done with just an abstract category. This is not possible. The data involved is not just a catego... | 3 | https://mathoverflow.net/users/4177 | 96169 | 56,246 |
https://mathoverflow.net/questions/96172 | 7 | Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set?
(This is mostly an idle question, but also motivated by the fact that it is a theorem of Milnor that a similar construction with the f... | https://mathoverflow.net/users/344 | What is the homotopy type of a free simplicial ring? | Let $R[-]$ be the free $R$-module functor, from sets to $R$-modules, and $T\_R$ the free (tensor) $R$-algebra functor, from $R$-modules to $R$-algebras. The free $R$ algebra functor from sets to $R$-algebras is the compoisite $T\_RR[-]$.
Given a simplicial set $X$, the homotopy type of $R[X]$ is well understood. It i... | 9 | https://mathoverflow.net/users/12166 | 96195 | 56,255 |
https://mathoverflow.net/questions/96158 | 10 | Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2\_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2\_k$. To be precise, does there exist a birational isomorphism $F:\mathbb{P}^2\_k \to \mathbb{P}^2\_k$ such that $F$ and $f$ agree on a (non empty) open subs... | https://mathoverflow.net/users/23501 | Extending birational isomorphisms between planar curves to the P^2 | In general the answer is **no**.
This kind of question is studied in more generality in the paper by Mella and Polastri "Equivalent birational embeddings", Bull Lond. Math. Soc. **41** (2009) 89-93, [http://arxiv.org/abs/0906.4858.](http://arxiv.org/abs/0906.4858)
They prove that two birational embeddings of $X$ in... | 12 | https://mathoverflow.net/users/7460 | 96196 | 56,256 |
https://mathoverflow.net/questions/96211 | 8 | **Question:** is there any example of a finitely presented (or at least finitely generated) group $G$ with an infinite cyclic subgroup $C \leqslant G$ such that the word problem in $G$ is solvable but the membership problem for $C$ in $G$ is unsolvable?
I would guess that such examples should exist, since otherwise a... | https://mathoverflow.net/users/7644 | Membership problem for cyclic subgroups | @Ashot: Finitely presented examples exist. It is in our paper with Olshanskii, Olshanskii, A. Yu.; Sapir, M. V., Length functions on subgroups in finitely presented groups. Groups—Korea '98 (Pusan), 297–304, de Gruyter, Berlin, 2000.
An easier construction is this. Take the free Abelian group $A$ with generators $x\... | 8 | https://mathoverflow.net/users/nan | 96213 | 56,262 |
https://mathoverflow.net/questions/96194 | 2 | Consider two finite point sets $P$ and $Q$ from (the same) Euclidean space, assume that they have the same cardinality $n$ and fix a bijection $\phi:P\to Q$. Define an undirected bipartite graph $G\_\phi$ where each $p \in P$ is linked via a single edge to $\phi(p) \in Q$ and this edge has weight $\|p - \phi(p)\|$.
W... | https://mathoverflow.net/users/18263 | Interpolating Bijections of Point Sets in Euclidean Space | Impose the condition that the functions $\phi\_{ij}$ are closed under composition: $\phi\_{bc} \circ \phi\_{ab} = \phi\_{ac}$. Then the nonstrict version of your inequality holds with $\alpha = \frac{k-1}{k}$. To prove it, enumerate $P\_1 = p\_{11},...,p\_{1N}$, say, then push this enumeration around by the $\phi$'s to... | 2 | https://mathoverflow.net/users/20787 | 96215 | 56,263 |
https://mathoverflow.net/questions/96207 | 1 | As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number.
$$
\psi(z)=\ln z+\sum\_{n=1}^{\infty}{\frac{B\_n}{nz^n}}
$$
$B\_n$ is the first Bernoulli numbers.
How to make a proof?
| https://mathoverflow.net/users/22954 | An asymptotic series for the digamma function | We can prove this, using Euler-Maclaurin Formula.
Here is a introduction from Wikipedia.
<http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula>
This is a quite easy problem.
To Admin,
You may be able to consider deleting this question, thanks. ^\_^
| 2 | https://mathoverflow.net/users/22954 | 96218 | 56,265 |
https://mathoverflow.net/questions/96185 | 9 | In A. Fröhlich's article *Local Fields* in *Algebraic Number Theory*, the following claim is made: if $R$ is a Dedekind domain with field of fractions $K$, $L$ is a finite separable extension of $K$ and $S$ is the integral closure of $R$ in $L$, and $x$ is an element of $S$ with minimal polynomial $g$, then, "by Euler'... | https://mathoverflow.net/users/6779 | Which formulae of Euler is Fröhlich referring to? | I believe the reference is to this formula of Euler (see [here](http://tinyurl.com/bw2a9gm)): If $P(x)/Q(x)$ is a rational function and $ax+b$ is a simple factor of $Q(x)$, then the coefficient of $1/(ax+b)$ in the partial fraction decomposition of $P/Q$ is given by
$$ \lim\_{x\to \frac{-b}{a}} \frac{a P(x)}{Q'(x)}. $$... | 5 | https://mathoverflow.net/users/430 | 96220 | 56,266 |
https://mathoverflow.net/questions/96192 | 2 | $K=Q(\sqrt{d} ) , d<0 $, $\Gamma $ an arithmetic subgroup of $G=SU(2,1)(K)$ . Is $\cup\_{g\in G}(g^{-1}\Gamma g)$ dense in G for the complex topology?
| https://mathoverflow.net/users/3945 | density of conjugate of arithmetic subgroup | Since $SU(2,1)(K)$ is dense in $H=SU(2,1)({\mathbb C})$ (as in your
[previous question](https://mathoverflow.net/questions/95833/density-in-su2-1)), it suffices to consider density in $H$ of the union of $H$-conjugates of your arithmetic subgroup $\Gamma$. Now, I will think of $H$ as a *real* algebraic Lie group. All ... | 3 | https://mathoverflow.net/users/21684 | 96221 | 56,267 |
https://mathoverflow.net/questions/96182 | 7 | In the theory of action-angle variables, you wind up having to solve integrals with a characteristic square-root behavior near the endpoints to express the action in terms of the orbital quantities. For the Kepler potential, this integral is
$$
I = \int\_{a}^{b}{\sqrt{\left(r - a\right)\left(b - r\right)} \over r}\,{\r... | https://mathoverflow.net/users/23504 | Action Integral | There are multiple ways of going about the evaluation of this integral and different methods are appropriate depending on what you need and how you intend to generalize the integrand.
If you are only interested in the numerical value for fixed parameters $a$ and $b$. Then this integral can be handled essentially "out... | 9 | https://mathoverflow.net/users/2622 | 96227 | 56,268 |
https://mathoverflow.net/questions/96204 | 6 | I began with problem which looked simple in the beginning but became increasingly complex as I dug deeper.
**Main questions**: Find the number of solutions $s(n)$ of the equation
$$
n = \frac{k\_1}{1} + \frac{k\_2}{2} + \ldots + \frac{k\_n}{n}
$$
where $k\_i \ge 0$ is a non-negative integer. This is my main question... | https://mathoverflow.net/users/23388 | A simple looking problem in partitions that became increasingly complex | I've got the following counts (which agrees with Brendan's):
1: 1
2: 3
3: 10
4: 55
5: 196
6: 2730
7: 10032
8: 108999
9: 973258
10: 20780331
11: 79309308
12: 2614200602
13: 10073335754
14: 288845706742
15: 11805287917646
16: 254331289285523
| 8 | https://mathoverflow.net/users/7076 | 96228 | 56,269 |
https://mathoverflow.net/questions/96225 | 4 | Let $F = \langle a,b \rangle$ be a non-abelian free group.
>
> **Question:** Is there an algorithm that takes as input $x,y,z \in F$ and answers the question whether $x$ is a product of conjugates of $y$ and $z$, i.e. whether there exists $g,h \in F$ with
> $$ x = gyg^{-1} \cdot hzh^{-1} ?$$
>
>
>
It is obvio... | https://mathoverflow.net/users/8176 | Extension of conjugacy problem | Yes, it is an equation in the free group. There exists an algorithm solving any equation due to Makanin, Makanin, G. S.
Equations in a free group.
Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1199–1273.
| 10 | https://mathoverflow.net/users/nan | 96231 | 56,271 |
https://mathoverflow.net/questions/96190 | 7 | I know that the above doesn't exist, but do bear with me. I need estimates/formulas for entries of certain eigenvectors and Cramer's rule keeps popping up in my mind. So, what can play an anlogous role in this case?
| https://mathoverflow.net/users/22051 | Cramer's rule for eigenvectors | If $u$ is a unit eigenvector with eigenvalue $\lambda$ of a Hermitian matrix
$$ A\_n = \begin{pmatrix} a & X^\* \\\\ X & A\_{n-1} \end{pmatrix}$$
with $a$ a real number, $X$ an $n-1$-dimensional row vector, and $A\_{n-1}$ an $n-1 \times n-1$ Hermitian matrix, then (provided that $\lambda$ is not an eigenvalue of $A\_{n... | 17 | https://mathoverflow.net/users/766 | 96235 | 56,274 |
https://mathoverflow.net/questions/96239 | 3 | I am faced with the following problem :
Originally (at time 0) I have a number of data samples $x^0\_{1...n}$ (normalised : $E[x] = 0, Var[x] = 1$) from which I have calculated the covariance matrix $C^0 = X^T X$ (where $X$ is the matrix of data samples), and the corresponding determinant $|C^0|$ (I could also store ... | https://mathoverflow.net/users/23514 | Determinant of an updated Covariance matrix | First compute a Cholesky factorization of the covariance matrix. Now your tentative new covariance matrix is a rank-2 update of the old,
$$
M\_{new}=M\_{old}+\frac{1}{n}x\_{new}x\_{new}^T-\frac{1}{n}x\_{old}x\_{old}^T.
$$
You can use Sylvester's formula [here](https://en.wikipedia.org/wiki/Matrix_determinant#Sylvester... | 4 | https://mathoverflow.net/users/1898 | 96240 | 56,277 |
https://mathoverflow.net/questions/96222 | 9 | Let $\mathcal{I}$ be an uncountable set. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, and $E\_i, i\in \mathcal{I}$ be a measurable set such that $\mathbb{P}(E\_i)=1$. What can we say about $\mathbb{P}(\cap\_{i\in \mathcal{I}} E\_i)?$.
I know that an uncountable intersection of measurable sets is no... | https://mathoverflow.net/users/12586 | Intersection of an uncountable number of sets. | If the uncountable number of sets that you're intersecting is small enough, you *might* be able to guarantee that the intersection is measurable (and in fact of measure 1) --- it depends on information about the set-theoretic universe that the usual axioms (ZFC) don't decide. Specifically, the *additivity of measure* i... | 7 | https://mathoverflow.net/users/6794 | 96245 | 56,279 |
https://mathoverflow.net/questions/96223 | 4 | Is this identity or an equivalent one already referenced in the litterature? In particular, is it even true?
${\frac{\left ( -1 ; e^{-4\pi} \right) ^2\_{\infty}}{\left ( e^{-2\pi} ; e^{-2\pi} \right) ^4\_{\infty}} = \frac{32 \pi^3 \left(\sqrt{2}-1 \right )\sqrt{2^\frac{1}{4}+2^\frac{3}{4}}}{\Gamma^4{\left(\frac{1}{4}... | https://mathoverflow.net/users/22279 | q-Pochhammer Symbol Identity | The equality is indeed correct. It follows from identities in Ramanujan's notebook.
First notice that
$\left(-1;e^{-4\pi}\right)\_{\infty}^{2}=2\left(-e^{-4\pi};e^{-4\pi}\right)\_{\infty}^{2},$
so we are trying to prove the identity
$$\frac{\left(-e^{-4\pi};e^{-4\pi}\right)\_{\infty}^{2}}{\left(e^{-2\pi};e^{-2\pi... | 10 | https://mathoverflow.net/users/12176 | 96249 | 56,281 |
https://mathoverflow.net/questions/96243 | 4 | The circle packing theorem (Koebe–Andreev–Thurston theorem) states that every finite planar graph is the nerve of some disk packing in the plane, where the nerve of a packing $P$ is a graph $G=(V,E)$, where there is an edge $(u,w) \in E(G)$ when $\partial P\_{u} \cap \partial P\_{w} \neq \varnothing$, for disks $P\_{u}... | https://mathoverflow.net/users/20343 | Generalizing the circle packing theorem to 3-dimensions | @Lee's comment is correct, and the answer to the question he cites give an almost complete picture, but you might also want to look at the following:
Combinatorial scalar curvature and rigidity of ball packings
D. Cooper and I. Rivin
Math Res Letters, 1996
(note: all the results in the papers are correct, but one ... | 3 | https://mathoverflow.net/users/11142 | 96251 | 56,282 |
https://mathoverflow.net/questions/96255 | 3 | Does taking direct summands/sums preserve formality of ext-algebras? More precisely:
Given an abelian category, say linear over a field and with enough injectives, one gets an $A\_\infty$-srutcture on the $ext$-algebras of its objects. Let $X,Y$ be objects of our category.
What is the relation between the following a... | https://mathoverflow.net/users/2837 | Formality of Ext algebras and direct sums | My understanding is that formality of the DGA $Ext^\bullet(X\oplus Y,X\oplus Y)$ implies 1), but also formality of $Ext^\bullet(X,Y)$ and $Ext^\bullet(Y,X)$ as bimodules over $Ext^\bullet(X,X)$ and $Ext^\bullet(Y,Y)$, and that this is a much stronger condition.
For instance, let $(E,p)$ be an elliptic curve over a fi... | 8 | https://mathoverflow.net/users/2356 | 96258 | 56,285 |
https://mathoverflow.net/questions/95647 | 3 | We know that Calabi constructed some extremal metrics on $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$ which are not constant scalar curvature ones.
I just want to know given a Kähler class in $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$ ($n\geq 3$), is there any criterion to detemine whether or not there contains a constan... | https://mathoverflow.net/users/19071 | A question about the existence of a constant scalar curvature metric on $\mathbb{C}P^n\#\overline{\mathbb{C}P^n}$ | There are no constant scalar curvature Kähler metrics on $M$, the blowup of $\mathbb{CP}^n$ at one point in any Kähler class and for any $n>1$.
This is because of the Lichnerowicz-Matsushima obstruction, which in this case says that if any such metric existed then $Aut^0(M)$ the connected component of the identity o... | 6 | https://mathoverflow.net/users/13168 | 96261 | 56,287 |
https://mathoverflow.net/questions/96246 | 5 | I'm looking for a reference for the following result, which is a generalization of the classical theorem of Dirichlet on the approximability of real irrationals by rational numbers:
Let $k$ be a number field, $O$ its ring of integers, $v$ an infinite place of $k$, $\alpha$ any element of the completion $k\_v$. Let $\... | https://mathoverflow.net/users/23519 | Reference wanted for generalized Dirichlet's Theorem | It seems like this book has the reference you need.
Schmidt, Wolfgang M. Diophantine approximation. Lecture Notes in Mathematics, 785. Springer, Berlin, 1980.
Chapter VIII, Theorem 2A.
It's available on the Springlink website.
Hope it helps.
| 4 | https://mathoverflow.net/users/22095 | 96265 | 56,290 |
https://mathoverflow.net/questions/96285 | 9 | So it was conjectured that if all elliptic curves over $\mathbb{Q}$ are ordered by their heights, then the average rank is $\frac{1}{2}$.
Brummer initially showed assuming BSD and GRH that the average rank is bounded by 2.3. Since then many improvements have been made. In my search, I found the slides for a talk by ... | https://mathoverflow.net/users/22095 | Average rank of elliptic curves over $\mathbb{Q}$ | Color me surprised if people no longer believe that the distribution of elliptic curves is half rank zero, half rank one, and a density zero subset of higher rank curves. To my knowledge this is still a conjecture that people believe.
I believe the state of the art results are still due to Bhargava and Shankar, and a... | 11 | https://mathoverflow.net/users/3384 | 96287 | 56,303 |
https://mathoverflow.net/questions/96292 | 13 | I would like prove that the following diophantine equation is unsolvable: $m!+27=n^3$.
Thanks in advance.
| https://mathoverflow.net/users/23528 | A hard diophantine equation: $m!+27=n^3$ | I am happy to report that the equation has no solution. I kept my original response, and put the remaining arguments in the "EDIT" section below.
Here is a quick proof that there are only finitely many solutions.
We use $m!=(n-3)(n^2+3n+9)$. Here $n$ is divisible by $3$, hence $n^2+3n+9$ is not divisible by any pri... | 34 | https://mathoverflow.net/users/11919 | 96294 | 56,305 |
https://mathoverflow.net/questions/96146 | 25 | Does there exist a closed smooth manifold $M$ and a diffeomorphism $f\colon M \to M$ such that:
1. $f$ is isotopic to the identity,
2. $f$ is of finite order, $f^n=ID$, and
3. $f$ is not contained in the image of any circle action, $S^1\to \operatorname{Diff}(M)$?
If the answer is yes, what is an example?
| https://mathoverflow.net/users/23500 | Diffeomorphisms of finite order not in the image of a circle action | Such examples exist in dimension 5, they are contained in the paper by Cameron Gordon "On the higher-dimensional Smith conjecture", Proc. London Math. Soc. (3) 29 (1974), 98–110. Namely, Gordon proves in Theorem 5 of this paper that (for every $n\ge 5$) there are infinitely many smooth knots $K=S^{n-2}\subset S^n$ so t... | 19 | https://mathoverflow.net/users/21684 | 96301 | 56,308 |
https://mathoverflow.net/questions/96289 | 12 | In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, Diamond, and Taylor.
I'm asking this as a matter of interest. Are there any other fields over which elliptic curves are kn... | https://mathoverflow.net/users/22095 | Extensions of the modularity theorem | Yes, this is a *very* active area -- one of the major themes of current research in number theory.
Much of the recent work has focussed on proving something slightly weaker, but easier to get at, than modularity. An elliptic curve $E$ over a number field $K$ is said to be *potentially modular* if there is a finite e... | 16 | https://mathoverflow.net/users/2481 | 96304 | 56,310 |
https://mathoverflow.net/questions/96296 | 1 | Let $X$ and $Y$ be two proper varieties of dimension $n$ over a field $k$. I'm looking for "reasonable" conditions, under which, there exists a proper and dominant morphism $f:V\to \mathbb{P}^1\_k$, with $V$ a variety of dimension $n+1$, such that $X=f^{-1}(0)$ and $Y=f^{-1}(\infty)$.
| https://mathoverflow.net/users/9516 | Putting two complete varieties in a family over the projective line | I will assume that by "variety" you mean (at least) "irreducible", otherwise as *Daniel Litt* points out the question is trivial.
Further, let's assume that $V$ is projective (and hence necessarily $X$ and $Y$), as *ulrich* points out, it will already be difficult enough.
Consider a common polarization of $X$ and $... | 4 | https://mathoverflow.net/users/10076 | 96309 | 56,313 |
https://mathoverflow.net/questions/96312 | 0 | Let $(M,\tau)$ be a finite von Neumann factor (in my case $M=R^\omega$, but I don't think this additional hypothesis might be useful for this particular problem) and fix a projection $p$. Let $\tau\_p$ (resp. $\tau\_{1-p}$) denote the unique normalized trace on $pMp$ (resp. $(1-p)M(1-p)$.
**Convention.** Given two pr... | https://mathoverflow.net/users/13809 | Geometric approximation of projections in a finite von Neumann factor | The point here is that you can distribute the sum over any function $f$:
This is almost obvious for the elementary operations:
Let $p\_1,p\_2\leq p$ and $q\_1,q\_2\leq 1-p$, then we compute that
$(p\_1+q\_1)\wedge (p\_2+q\_2)=(p\_1\wedge p\_2)+(p\_1\wedge q\_2)+(q\_1\wedge p\_2)+(q\_1\wedge q\_2)=(p\_1\wedge p\_2)+(q... | 2 | https://mathoverflow.net/users/2055 | 96317 | 56,317 |
https://mathoverflow.net/questions/96318 | 5 | I was reading Dominic Joycee [article](http://arxiv.org/abs/0910.3518) on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here i moved to Melrose unpublished book on Manifold with corner.
Is the theory for Complex manifold with bound... | https://mathoverflow.net/users/23534 | complex manifold with corner | There are some related results about compact Stein 4-manifolds with boundary as Lefschetz fibrations over the disk (whose fiber has non-empty boundary). Corners in this case arise naturally on the total space. References includes Loi-Piergallini's theorem, and subequent works of Akbulut-Ozbagci (simply google with thes... | 2 | https://mathoverflow.net/users/23193 | 96321 | 56,319 |
https://mathoverflow.net/questions/96324 | 2 | Given a smooth projective variety $X / \mathbb F\_p$, can one find a smooth projective $\mathcal X / \mathbb Z\_p$ such that $\mathcal X \times\_{\mathbb Z\_p} \mathbb F\_p = X$? (or similarly with $\mathbb Z$ instead of $\mathbb Z\_p$).
Thank you!or
| https://mathoverflow.net/users/18116 | Lift smooth projective varieties over $\mathbb F_p$ to $\mathbb Z_p$? | Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}\_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}\_p$ other than $\mathbf{P}\_1$.
**Addendum** A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize... | 11 | https://mathoverflow.net/users/2821 | 96329 | 56,322 |
https://mathoverflow.net/questions/96315 | 2 | I was trying to construct some element with specific properties in an ultraproduct and it boils down to a question which seems relatively natural but leaves me perfectly clueless.
$\textbf{Question:}$ Pick some $\alpha \in (0,1)$. Let $\phi:\mathcal{P} \_{fin}(\mathbb{N}) \to \mathcal{P} \_{fin}(\mathbb{N})$ be any f... | https://mathoverflow.net/users/18974 | Ultrafilter and contracting maps | The answer to the original question (without the "increasing" requirement) is no, for any $\alpha<1$. Given $\alpha<1$, fix a positive integer $n$ so large that $\alpha+(1/n)<1$. Then, for each finite set $S\subseteq\mathbb N$, define $\phi(S)\subseteq S$ with $|\phi(S)|\geq\alpha|S|$ in such a way that $\phi(S)$ avoid... | 4 | https://mathoverflow.net/users/6794 | 96333 | 56,324 |
https://mathoverflow.net/questions/96320 | 2 | is any strongly regular graph a regular two-graph?
two-graph:a two graph is a collection $B$ of 3-subsets a set $X$ with the property that, for any 4-subset $Y$ of $X$, an even numbers of $B$ belong to $Y$.
regular two-graph:a two-graph is regular if it is a 2-design (with parameters $2-(n,3,\lambda)$ for some $\la... | https://mathoverflow.net/users/22967 | strongly regular graph as two-graph | No, there is a correspondence between certain strongly regular graphs and two-graphs but those strongly regular graphs have specific and restricted parameters.
| 6 | https://mathoverflow.net/users/1492 | 96337 | 56,325 |
https://mathoverflow.net/questions/96338 | 6 | Let $f(x,y)$ be some bounded with its derivatives continuous function on $\Omega \times \overline{\Omega}$, where $\Omega$ is a domain in $\mathbb{R}^n$. Let $f(\,\,\cdot\,,\,y) \in \mathcal{E}(\Omega)$ for any fixed $y$. Let $L\_{x} \in \mathcal{E}'(\Omega)$. Is it true that
$$\int\limits\_{\overline{\Omega}} L\_{x}... | https://mathoverflow.net/users/17896 | Integration under functional sign | Suppose first that $L\_x\in C^\infty\_0(\Omega)$. Then the equality you ask about is Fubini's theorem.
Suppose now that $L\_x$ is not necessarily smooth. Choose a sequence $\newcommand{\ve}{\varepsilon}$ $L\_{\ve,x}\in \mathscr{E}'(\Omega)$ that converges to $L\_x$ in the weak sense. Then one needs to prove that
$... | 6 | https://mathoverflow.net/users/20302 | 96348 | 56,328 |
https://mathoverflow.net/questions/96290 | 11 | This is probably an easy question for the experts:
Given two rational functions $f$, $g$ on a non-singular projective algebraic curve X (over an algebraically closed field $k$) and $p \in X$, one defines the Weil symbol $(f, g)\_p$ as
the value of $(-1)^{ab} f^a g^{-b}$ at $p$ where $a = v\_p(g)$ and $b = v\_p(f)$. ... | https://mathoverflow.net/users/21491 | Weil reciprocity vs Artin reciprocity | Weil reciprocity actually holds for arbitrary fields $k$, not necessarily algebraically closed: just let $x$ run over all closed points of $X$, and replace your expression for $(f,g)\_x$ by its norm from $k(x)^\* $ down to $k^\*$.
Then the connection with Artin reciprocity occurs when $k$ is a finite field (of size s... | 8 | https://mathoverflow.net/users/3931 | 96349 | 56,329 |
https://mathoverflow.net/questions/96191 | 5 | I am looking for a reference concerning operator theoretical Models of $K(\mathbb{Z},3)$. Stolz-Teichner briefly say in "what is an elliptic object" that a certain hyperfinite Type III-factor, called "local fermions on the circle" should be the right thing. Thanks
| https://mathoverflow.net/users/23506 | Operator Theoretical Models for $K(\mathbb{Z}, 3)$ | Here is a $C^\*$-algebraic version of the model described in Andre Henriques' answer (the latter was linked by David Corfield in the accepted answer above):
Let $\mathcal{O}\_2$ be the Cuntz algebra generated by two partial isometries $s\_1$ and $s\_2$ subject to the relations $s\_i^\*s\_j = \delta\_{i,j}$ and $s\_1... | 7 | https://mathoverflow.net/users/3995 | 96357 | 56,333 |
https://mathoverflow.net/questions/96343 | 7 | I just started to learn about the Ricci flow and try to understand the Ricci flow evolution equation. It states that a one-parameter family $g\_t$, $t\in[0,T)$ of Riemannian metrics on a smooth closed manifold $M$ is a solution of the equation
$$ \frac{\partial g\_t}{\partial t}=-2 Ric\_{g\_t}. $$
But, in order to be a... | https://mathoverflow.net/users/23509 | the left hand side of the Ricci flow equation at the initial value | Misha's comment could be a bit misleading. In particular, it is not true that the Ricci flow should exist on a slightly bigger interval $(-\epsilon,T)$ with $g(0) = g\_0$. One way to see this is by thinking about a theorem of Bando (see <http://www.springerlink.com/content/v0764574t4764138/> if you have access) which s... | 8 | https://mathoverflow.net/users/1540 | 96363 | 56,336 |
https://mathoverflow.net/questions/96314 | 9 | In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level structure corresponds to higher torsion points (when we say "higher" we mean, e.g., $E[n]$ is "higher" than $E[m]$ if $m \mid n$). In compl... | https://mathoverflow.net/users/1355 | Adelic formulations of complex multiplication and modular curves | This is an expansion of my comment above: You can view the modular curve of level $K$ over $\mathbb{C}$ as the double coset space
$$GL\_2(\mathbb{Q})\backslash\mathbb{H}^{\pm}\times GL\_2(\mathbb{A}\_f)/K.$$
Here, $\mathbb{H}^{\pm}$ is the union of the half planes. This set is in bijection with isogeny classes of ellip... | 11 | https://mathoverflow.net/users/7868 | 96384 | 56,346 |
https://mathoverflow.net/questions/96234 | 6 | Let $(M^n,g)$ be a Riemannian manifold and let $W$ be its Weyl tensor. For a given ONB, does the identity
$$W\_{ijkl}W\_{ijkm}=\frac{1}{n}|W|^2g\_{lm}$$
hold? I think I've seen it somewhere but I'm not sure whether this is valid only in dimension four (in this case, this is certainly true).
| https://mathoverflow.net/users/20823 | Identity of the Weyl-Tensor | This does not hold for $n>4$. To see this, start with a Ricci-flat 4-manifold $N^4$ that is not flat, and let $M^{n}= N^4 \times \mathbb{R}^{n-4}$, endowed with the product metric. Then the metric on $M$ is Ricci-flat, so its Riemann curvature tensor is its Weyl tensor and $W\_{ijkl}=0$ when any index is greater than $... | 8 | https://mathoverflow.net/users/13972 | 96389 | 56,349 |
https://mathoverflow.net/questions/96341 | 4 | recently, I am reading Tomek Bartoszynski's book"Set Theory On The Structure Of The Real Line"
There is a lemma I don't understand it's proof.(P244 Lamma 4.6.10),it's original expression as follows.
Suppose that $P$ is a forcing satisfying the laver condition,let $<\dot{a\_n}:n\in\omega >$
be an $M\star P-name$ fo... | https://mathoverflow.net/users/22161 | a result about Laver property | I think you are correct that the Mathias forcing is a red herring, the same lemma should apply to any ground model not just a Mathias extension. (The lemma was placed in the middle of a proof in the text, so I can see why they might not have been concerned with the extra generality). It also looks like the proof is, if... | 3 | https://mathoverflow.net/users/2436 | 96392 | 56,351 |
https://mathoverflow.net/questions/96378 | 29 |
>
> Let $M$ be a compact complex connected [but *not necessarily kähler*] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^\*)^n \to M$$ such that the image is open. Is the image necessarily dense in $M$?
>
>
>
Motivation: My intuition (which comes from the algebraic world) says that the answer ... | https://mathoverflow.net/users/5094 | Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense | An example exists already for $M={\mathbb C}P^2$, furthermore, there exists an injective holomorphic map $f: {\mathbb C}^2\to {\mathbb C}^2\subset {\mathbb C}P^2$ whose image is open but not dense. Recall that a domain $\Omega$ in ${\mathbb C}^2$ is called a *Fatou-Bieberbach* (FB) domain if $\Omega\ne {\mathbb C}^2$ a... | 36 | https://mathoverflow.net/users/21684 | 96395 | 56,352 |
https://mathoverflow.net/questions/96381 | 3 | A sort of continuation of [Comparing distributions with moments](https://mathoverflow.net/questions/96165/comparing-distributions-with-moments)
Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\log n),$$ with $\beta > 0$. Can I conc... | https://mathoverflow.net/users/8537 | Bounds on tails with moments | You need a weaker conjecture, since the cdf $(1+x)e^{-x}$ for $x\ge 0$ has moments of the desired form yet it isn't like $1-Ae^{-cx}$. More generally, the distribution with cdf $(1+x^k)e^{-x}$ has moments with logarithm
$$n \ln n - n + (1/2+k)\ln n + O(1) .$$ Try cdf $\exp(-x+x^{1/2})$.
| 2 | https://mathoverflow.net/users/9025 | 96406 | 56,359 |
https://mathoverflow.net/questions/96404 | 8 | I'm looking for good surveys about characteristic classes of flat real vector bundles. Letting $G$ be $\text{SL}\_n(\mathbb{R})$ with the **discrete** topology, orientable flat $n$-dimensional real vector bundles are classified by $BG$, so the characteristic classes I'm looking for are elements of the cohomology of $BG... | https://mathoverflow.net/users/23558 | References/surveys concerning characteristic classes of flat vector bundles | I am not aware of much recent activity (after 2001, when Morita wrote his book) in this field, so I think it is still very valuable. Chern-Simons theory means something rather different today. The useful information on characteristic classes of flat vector bundles (CCFVB) seems to be scattered throughout the literature... | 6 | https://mathoverflow.net/users/9928 | 96417 | 56,362 |
https://mathoverflow.net/questions/96421 | 3 | Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves,
there exists a surjective morphism from $X\_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and morphisms from $X\_0(N)$ $\rightarrow$ $E$. One such $N$ is the conductor of $E$.
If there exists an elliptic curve $E$ a... | https://mathoverflow.net/users/20754 | Conductor of an elliptic curve | Yes. If there is a surjection $X\_0(N) \to E$ then the conductor of $E$ must divide $N$. Since there is no elliptic curve of conductor 1, an elliptic curve uniformized by $X\_0(p)$ for $p$ prime must have conductor $p$.
| 3 | https://mathoverflow.net/users/2481 | 96433 | 56,366 |
https://mathoverflow.net/questions/96356 | 6 | Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields:
* Type I: totally real, trivial involution
* Type II and III: quaternion algebras over totally real number fields
* Type IV: center is a CM field for which the restriction of the involuti... | https://mathoverflow.net/users/nan | Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields | See <http://www.math.nyu.edu/~tschinke/books/finite-fields/final/05_oort.pdf>, Section 15:
Only Type III and IV can occur, and III only for dimension $1$ or $2$.
| 2 | https://mathoverflow.net/users/nan | 96435 | 56,367 |
https://mathoverflow.net/questions/96429 | 2 | Every now and then I wonder what the official name of the relation $\sim$ between two vertices in a graph $G$ is that are mapped to each other by a graph automorphism, i.e. which are "structurally indistinguishable":
$$x \sim y \quad\text{iff}\quad (\exists g \in \text{Aut}(G))\ x = g(y)$$
I use to call such vertic... | https://mathoverflow.net/users/2672 | Name of vertices in the same orbit | In pretty much all texts or papers on graph theory that I've seen two vertices in the same orbit of the automorphism group are called *similar*. If $G-\lbrace u\rbrace\cong G-\lbrace v\rbrace$ but $u$ and $v$ are not similar, one calls such vertices *pseudosimilar*. The same terminology is used for edges. You will find... | 7 | https://mathoverflow.net/users/2384 | 96437 | 56,369 |
https://mathoverflow.net/questions/96419 | 4 | Given two path algebras $A$ and $B$, for example, A=: $1\to2$ B=: $1\to2\to3$, is the tensor product of A and B over a field $k$ a
path algebra? if yes, how to represent it by a quiver? also, how to construct $(A,B)$-bimodules?
| https://mathoverflow.net/users/21505 | The tensor product of two path algebras | It can be realized as a quotient of the Cartesian product of path algebras in question. You have to mod out by some commutativity relations.
| 1 | https://mathoverflow.net/users/12637 | 96439 | 56,370 |
https://mathoverflow.net/questions/96414 | 1 | Is there any necessary and sufficient condition for function $f$ such that:
$f(x)=\sum\_{k=1}^{\infty} f\_k(x)$ for all $x \in \mathbb{R}$,where $(f\_n )\_{n=1}^{\infty}$ is a sequence of periodic function on $\mathbb{R}$ ??
(note that $f\_n$ may not be integrable or measurable)
Besides,it is known that $ \lim\_{... | https://mathoverflow.net/users/22907 | represented as a series of periodic function | Every function can be written as the pointwise sum of a sequence of periodic functions.
Given $f$. Let $f\_1 = f$ on $(-2,2]$ and extend as a function of period 4. And follow the following recursive definition:
Let $ f\_{k+1}(x) = f(x) - \sum\_{n= 1}^k f\_k(x)$ for $x\in (-2^{k+1},2^{k+1}]$ and extend as a functio... | 1 | https://mathoverflow.net/users/3948 | 96442 | 56,373 |
https://mathoverflow.net/questions/96426 | 1 | Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$.
The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$.
Edit: I am looking for a proof. Thanks!
| https://mathoverflow.net/users/20754 | Elliptic subfields of a function field | Elliptic subfields of $K(C)$ correspond to finite morphisms from $C$ to an elliptic curve, which in turn correspond to elliptic factors of the Jacobian of $C$. Thus you get $0, 2, \infty$ essential elliptic subfields according to the decomposition of $\mathrm{Jac}(C)$ : it can be simple or isogenous to a product of ell... | 5 | https://mathoverflow.net/users/6506 | 96444 | 56,375 |
https://mathoverflow.net/questions/88255 | 4 | Let $S=\{1,2,\dots,m+n-1\}$.
An $m\times n$ matrix($\in S^{m\times n}$) is called silver matrix if
(a) There is no same numbers in the row or column. (like latin square)
(b) {$i$ th row}$\cup${$i$ th column}=S for all $1\leq i\leq min(m,n)$
Does silver matrix exist for all $m\neq n$ ?
If this conjecture is tr... | https://mathoverflow.net/users/21335 | Generalized silver matrix (related to defining number) | Yes. Silver matrices exist for all $n=m$ when $n$ is even and for all $n\neq m$. Notice that the problem for square matrices is essentially problem 4 in the 1997 International Math Olympiad.
One basic construction one needs is a symmetric latin square. These exist for all orders and are essentially equivalent to edge... | 6 | https://mathoverflow.net/users/2384 | 96449 | 56,376 |
https://mathoverflow.net/questions/96427 | 2 | I have seen somewhere the following results related to Lyapunov equation:
Let $A\in \mathbb{R}^n$ be a stable matrix in the sense of having negative real part eigenvalues. Let $\Re\lambda()$ denote the real part of a eigenvalu of a matrix.
(1) A necessary and sufficient condition for $\Re\lambda(A) > \lambda\_1$ is... | https://mathoverflow.net/users/23540 | Eigenvalue estimation by Lyapunov's method | Hale, Ordinary Differential Equations, Lemma 1.5 in Chapter X. This refers to the 1980 edition. It is the chapter on Liapunov's direct method.
| 2 | https://mathoverflow.net/users/12120 | 96454 | 56,377 |
https://mathoverflow.net/questions/94996 | 4 | Consider the equation
$-\Delta u + Vu=f$,
on a closed manifold (or on a bounded domain with homogeneous Neumann condition). Here one can assume whatever integrability or smoothness conditions on $V$ and $f$ one likes. One can show that if $V$ and $f$ are both nonnegative and not identically zero, then the unique so... | https://mathoverflow.net/users/824 | Lower bound on the solution of a Schrödinger-type equation | Here are a couple of ideas:
1 This yields a lower bound, but it is not in terms of a norm of f:
Let $\lambda=\min(f/V)$. Then $u-\lambda$ satisifies
$$-\Delta(u-\lambda)+V(u-\lambda)=f-\lambda V\ge 0.$$
Hence $u\ge\lambda$ by the maximum principle.
2 Assume V is bounded, and let $V\_M$ be its maximum:
Let $v$ be ... | 4 | https://mathoverflow.net/users/12120 | 96458 | 56,379 |
https://mathoverflow.net/questions/96453 | 3 | I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function
toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the n variable saddlepoint method").
Is such thimble a holomorphic manifold? Maybe under more restrictive assumptions?
| https://mathoverflow.net/users/23261 | Are Lefschetz thimbles holomorphic manifolds? | If by holomorphic manifold you mean that it happens to be a complex manifold then the answer is surely "not always", because in the case when the total space is $\mathbf{C}^3$ and the function is $(z\_1,z\_2,z\_3)\mapsto z\_1^2+z\_2^2+z\_3^2$ then the thimble living over the positive real axis will be $\mathbf{R}^3$ (w... | 4 | https://mathoverflow.net/users/10839 | 96462 | 56,380 |
https://mathoverflow.net/questions/96469 | 5 | Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are there examples showing that this inequality is not a sufficient condition? Are there methods to prove that two given ele... | https://mathoverflow.net/users/39082 | Matrices generating non-discrete subgroups of SL(2,R) | For more than you ever dreamed possible on the subject, see Jane Gilman's
MR1290281 (97a:20082)
Gilman, Jane(1-RTG2)
Two-generator discrete subgroups of PSL(2,R). (English summary)
Mem. Amer. Math. Soc. 117 (1995), no. 561, x+204 pp.
20H10 (22E40 30F35)
For a more concise paper which will answer your question,... | 4 | https://mathoverflow.net/users/11142 | 96470 | 56,384 |
https://mathoverflow.net/questions/96451 | 5 | If $G$ and $H$ are groups, then the map $BG\vee BH\to B(G\ast H)$ is a weak equivalence by van Kampen's theorem. However the classifying space $BM$ of a monoid can have arbitrary homotopy type so higher homotopy groups are involved.
If $M$ and $N$ are monoids, is the map $BM\vee BN\to B(M\ast N)$ still a weak equival... | https://mathoverflow.net/users/6301 | Does the classifying space of monoids commute with wedge sum up to weak equivalence? | Here is a high-tech point of view.
The inclusion functor $\mathrm{Groups}\to \mathrm{Monoids}$, has a left adjoint $F\colon \mathrm{Monoids}\to\mathrm{Groups}$, which is the group completion functor. You know that
$$\pi\_1(BM) = FM.$$
The claim is that if we instead consider the *total left derived functor* $LF$ o... | 14 | https://mathoverflow.net/users/437 | 96478 | 56,391 |
https://mathoverflow.net/questions/96446 | 5 | Is there some sort of classification of finite groups $G$ such that for at least one $n$ the group $G$ admit a free *isometric* action on the standard sphere $S^n $of curvature 1? Are there some simple criteria that permit to check (in some particular cases) if a given group has such an action (for at least on $n$) or ... | https://mathoverflow.net/users/13441 | Finite groups admitting free isometric actions on round spheres | After all these comments, a possible answer to your question goes as follows.
If $n$ is even, then the only group that can act freely and isometrically on $S^n$ is $\mathbb{Z}/2$. One way to see this is as in Max's answer above, i.e. by looking at the behavior of eingenvalues of orthogonal matrices. Here is another ... | 3 | https://mathoverflow.net/users/5069 | 96485 | 56,393 |
https://mathoverflow.net/questions/96488 | 2 | Let $(a\_n)$, $(b\_n)$ be the Fourier coefficients of a periodic, locally integrable function $f: \mathbb{R} \rightarrow \mathbb{R}$. Assume that $n^m a\_n, n^m b\_n \rightarrow 0$ when $n \rightarrow \infty$. By the Weierstrass test, $f$ is of class $C^{m-2}$. Is $f^{(m-2)}$ absolutely continuous or differentiable alm... | https://mathoverflow.net/users/19795 | Function with Fourier coefficient of order $o(n^{-m})$ | Under these assumptions the function is in the Sobolev space $H^{m-1/2-\epsilon}$ for any $\epsilon>0$. This implies in particular that $f^{(m-1)}$ is in $L^p$ for any $p<\infty$.
| 4 | https://mathoverflow.net/users/12120 | 96492 | 56,398 |
https://mathoverflow.net/questions/96455 | 5 | Let $f: Y \to X$ be a birational morphism of projective varieties. Let $\mathcal{M}$ be a very ample invertible sheaf on $Y$. Suppose also that:
* $f^{-1}$ is defined away from a single point $x \in X$.
* $f\_\* \mathcal{O}\_Y = \mathcal{O}\_X$.
Two questions:
(1) If $V$ is a set of global sections of $\mathcal{M... | https://mathoverflow.net/users/2579 | Can the pushforward of a very ample invertible sheaf under a birational morphism be reflexive? | Hi Sue, I think it can be reflexive.
Let's consider $X = \text{Proj} k[x,y,u,v,t]/\langle xy - uv \rangle$. This has only an isolated singularity at $x=y=u=v=0$ (it's the simplest non-Q-factorial singularity I know of). Fix $U$ to be the regular locus of $X$.
**(Blow up a divisor:)** Set $\pi: Y \to X$ to be the bl... | 3 | https://mathoverflow.net/users/3521 | 96496 | 56,400 |
https://mathoverflow.net/questions/96497 | 4 | Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times\_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(x,y):\mathrm{Aff}/U \to \mathrm{Ens}$ is represented by an algebraic $U$-space, for every $U \in \mathrm{Aff}/S$ and al... | https://mathoverflow.net/users/18289 | Representability of the diagonal morphism of stacks | This follows immediately from the definitions, in particular from the fiber product of stacks (see loc. cit. (2.2.2)). For $V \in \mathrm{Aff}$ (everything over the base $S$), a $V$-point of $U \times\_{X \times X} X$ is a triple $(i,z,\alpha)$, where $i$ is a $V$-point of $U$, $z$ is a $V$-point of $X$ and $\alpha : (... | 5 | https://mathoverflow.net/users/2841 | 96503 | 56,402 |
https://mathoverflow.net/questions/96516 | 3 | In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence:
*It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$.*
I am wondering what is the current status of this conjecture.
Thank you,
J.
| https://mathoverflow.net/users/20746 | When an AW*-algebra is a W*-algebra | I don't know who solved this problem originally, but see J.D. Maitland Wright, Wild AW∗-factors and Kaplansky-Rickart algebras, *J. London Math. Soc.* 13 (1976), 83–89 for a construction of a family of AW\* factors. These have trivial, hence commutative, center.
| 7 | https://mathoverflow.net/users/23141 | 96522 | 56,409 |
https://mathoverflow.net/questions/96518 | 7 | The ring $\mathbb{Z}[2\cos(\frac{\pi}{k})]$ is known to be a Euclidean domain for $k=3,4,5$ and $6$, because in those cases $2\cos(\frac{\pi}{k}) = 1, \sqrt{2},$ the golden ratio $\phi$, and $\sqrt{3}$ respectively, and $\mathbb{Z}, \mathbb{Z}[\sqrt{2}], \mathbb{Z}[\phi]$, and $\mathbb{Z}[\sqrt{3}]$ are all known to be... | https://mathoverflow.net/users/23582 | Is $\mathbb{Z}[2\cos(\frac{\pi}{k})]$ a Euclidean domain? | There's certainly no reason to expect that it will *always* be a Euclidean domain, since (as Voloch points out) to be a Euclidean domain it must necessarily be a PID. On the other hand, the converse is (essentially) true in this case. Namely, a theorem of Weinberger implies that if $K/\mathbb{Q}$ is Galois, totally rea... | 9 | https://mathoverflow.net/users/23584 | 96523 | 56,410 |
https://mathoverflow.net/questions/96524 | 2 | I apologize if this is too elementary. The following identity arises in cluster algebra, where I'm trying to find an expression for cluster variables. Let $a,b$ be any nonnegative integers. Then there are nonnegative integers $d\_0,...,d\_{a+b}$ (depending on $a,b$, but not $X$) such that
$$
{X\choose a} {X\choose b} =... | https://mathoverflow.net/users/13693 | Reference request : an elementary product-sum formula for binomial coefficients | $\binom{X}{a}\binom{X}{b}$ is the number of ways to choose a subset of size $a$ and a subset of size $b$ from a set of size $X$. The union of these two subsets is a subset of size anywhere from $\text{max}(a, b)$ to $a + b$, so $d\_i$ is the number of different ways a subset of size $i$ can be realized as the union of ... | 8 | https://mathoverflow.net/users/290 | 96525 | 56,411 |
https://mathoverflow.net/questions/96542 | 7 | I am currently reading a paper by De Weger and one theorem in it proves a bound for the Tamagawa number of any elliptic curve defined over $\mathbb{Q}$.
I was wondering if anyone has any good references/texts that provide an exposition on the Tamagawa number of an elliptic curve as I was unable to find one in the Ar... | https://mathoverflow.net/users/22095 | Tamagawa Number of Elliptic Curves over $\mathbb{Q}$ | I think I would start with Weil "Adeles and algebraic groups", but if you are looking for something more specifically associated to elliptic curves, maybe this survey of Guido Kings:
<http://epub.uni-regensburg.de/13613/1/MP6.pdf> is a good starting point to see the connection between the Equivariant Tamagawa Number co... | 4 | https://mathoverflow.net/users/10400 | 96547 | 56,419 |
https://mathoverflow.net/questions/96551 | 1 | Define $Re\lambda\_{min}(A)$ to be the minimum of the real parts of the eigenvalues of a matrix $A$. Let $A,B\in \mathbb{R}^{n \times n}$ be two matrice such that $Re\lambda\_{min}(A)>0$, $Re\lambda\_{min}(B)>0$ and $Re\lambda\_{min}(A-B)\geq 0$. Is $Re\lambda\_{min}(A)\geq Re\lambda\_{min}(B)$ right? And how to prove.... | https://mathoverflow.net/users/23540 | Inequality regarding the smallest real part of eigenvales | Here is a $2 \times 2$ counterexample:
$A=\begin{bmatrix}10 & 19 \\\\ 8 & 16\end{bmatrix}$
$B=\begin{bmatrix}9 & 2 \\\\ 17 & 7\end{bmatrix}$
If you have tacit assumptions, share them and we'll see if the statement becomes true!
| 2 | https://mathoverflow.net/users/22051 | 96555 | 56,423 |
https://mathoverflow.net/questions/96390 | 11 | Who introduced the Strong Atiyah Conjecture?
Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bounded operator $l^2(G)^n\to l^2(G)^n$. Let $K$ be the kernel of that operator and $p$ be the orthogonal projection onto $K$. Let $\delta$ be... | https://mathoverflow.net/users/nan | Strong Atiyah conjecture | I received a message from Thomas Schick, which answers my question. The strong
Atiyah conjecture was introduced jointly by Lueck and Schick (this was also one of the alternatives in Wolfgang Lueck's email to me), and Thomas Schick is responsible for the super-strong one. He was a postdoc working with Wolfgang Lueck at... | 10 | https://mathoverflow.net/users/nan | 96560 | 56,425 |
https://mathoverflow.net/questions/96563 | 8 | Suppose $V := \bigoplus\_{i \in \mathbb{N}}V\_i$ and $W := \bigoplus\_{i \in \mathbb{N}}W\_i$ are $\mathbb{N}$-graded vector spaces. Then their graded tensor power is defined by
$V \bigotimes W := \oplus\_{n \in \mathbb{N}} \oplus\_{i+j=n} V\_i \otimes W\_j$ and there is a natural isomorphism $\sigma\_{V,W}: V \bigoti... | https://mathoverflow.net/users/21965 | Origin of the sign convention in the Tensor product of graded vector spaces | *Is this the only one or are there others defined by other sign conventions?*
As has already been mentioned - there are essentially two but the Koszul one is amongst other things, the only one which makes the tensor product of complexes a complex.
Since I have been interested in this topic of late, I would point yo... | 8 | https://mathoverflow.net/users/14352 | 96569 | 56,429 |
https://mathoverflow.net/questions/96566 | 3 | We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$.
one of the properties of these topological spaces is that each point of them is a $G\_\delta$-point.
there are a lot of interesting topological spaces which are not metrizab... | https://mathoverflow.net/users/23317 | Existence of a non-submetrizable topological space $(X, \tau)$ | The Alexandroff´s double arrow space (i.e. $[0,1] \times \{0,1\}$ with the lexicographic order) is first countable (so any point is a $G\_\delta$), but is compact and non-metrizable (hence it is not submetrizable).
| 6 | https://mathoverflow.net/users/17836 | 96570 | 56,430 |
https://mathoverflow.net/questions/96567 | 4 | If I have a homomorphism $f: G\to H$ of groups, I get a homomorphism $R(f)\colon R(H)\to R(G)$ of representation rings in the opposite direction, by composition. Given two homomorphisms $f\_1,f\_2\colon G\to H$, it is sometimes the case that $R(f\_1)=R(f\_2)$. For example, this happens whenever $f\_1$ and $f\_2$ are co... | https://mathoverflow.net/users/23598 | When do two maps between groups give the same map between representation rings? | Your two questions are not just closely related but identical. If $R(f\_1)=R(f\_2)$ then $\chi(f\_1(x))=\chi(f\_2(x))$ for all characters $\chi$, and standard representation theory allows you to deduce that $f\_1(x)$ is conjugate to $f\_2(x)$.
For a basic example where pointwise conjugacy is different from conjugacy,... | 9 | https://mathoverflow.net/users/10366 | 96571 | 56,431 |
https://mathoverflow.net/questions/96506 | 1 | While reading [Brylinski](http://books.google.com/books?id=ta5UB1D64_gC&printsec=frontcover&dq=brylinski+loop+spaces&hl=en&sa=X&ei=jtmqT724Nurl6QGF55SxBA&ved=0CDMQ6AEwAA#v=onepage&q=brylinski%2520loop%2520spaces&f=false) I am trying to understand the descent of morphisms of sheaves.
In trying to form a new definition... | https://mathoverflow.net/users/19926 | Descent of Morphisms of Sheaves | Yes, the property is satisfied (note that $\Gamma(Y,f^{-1}A)=A(f(Y))$, since $f$ is open). I didn't read the book, but I imagine that the point is not to give a strange definition of a sheaf on a topological space, but rather to motivate the generalisation to situations where you know what maps you wish to consider to ... | 3 | https://mathoverflow.net/users/10174 | 96574 | 56,433 |
https://mathoverflow.net/questions/96577 | 1 | Given a (orthogonal) basis $(\sigma\_n)\_{n=1,\dots,K}$ of the algebra $u(N)$, we can represent any element $U$ of the corresponding group $U(N)$ in the form
$U=e^{i\sum\_{n=1}^K\varphi\_n\sigma\_n}$.
Is it also possible to represent every element of $U(N)$ in the following form as a product of elementary (Givens) ... | https://mathoverflow.net/users/23602 | Representing elements of U(N) or SO(N) by elementary rotations exp(i phi_n sigma_n) | See [this paper](http://jmp.aip.org/resource/1/jmapaq/v47/i4/p043510_s1).
| 1 | https://mathoverflow.net/users/7410 | 96578 | 56,435 |
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