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https://mathoverflow.net/questions/77985 | 4 | In the definition of a locally connected space we demand every neighbourhood of a point to satisfy certain condition whereas for a locally compact space we demand that one neighbourhood be there with the required property.
Is there some reason for this difference? Is it so that a compact space needs(?) to be locally... | https://mathoverflow.net/users/9991 | locally connected versus locally compact | 1. One could phrase local compactness using neighbourhood bases as well (in the Hausdorff case, at least) if desired: once one has one precompact open neighbourhood, one automatically has a whole neighbourhood base of precompact sets, since any subset of a precompact set is still precompact. (And in practice, this is o... | 16 | https://mathoverflow.net/users/766 | 78034 | 47,072 |
https://mathoverflow.net/questions/77720 | 3 | Let $g(x,c)$ be a uniformly random integer in the range $(x,x+c)$ and $LCM[x\_1,x\_2...x\_i]$ the lowest common multiple of the integers $x\_i$.
**A)** Does the limit of (the asymptotics of $LCM[g(3^1,c),g(3^2,c),...,g(3^x,c)]$ as $x→∞$) as $c→∞$ exist?
**B)** How to find the limit of A) if it exists ?
**C)** Giv... | https://mathoverflow.net/users/18359 | About the asymptotics of LCM | There are some basic results about LCM that you can use. Here's one:
for any increasing sequence of positive integers $a\_1 \lt \ldots \lt a\_n$, one has
$a\_n \leq LCM(a\_1 , \ldots, a\_n) \leq a\_1 \times \ldots \times a\_n$ . So if $a\_n$ increases quickly enough, the difference between the logs of the upper and low... | 1 | https://mathoverflow.net/users/3402 | 78046 | 47,076 |
https://mathoverflow.net/questions/78043 | 2 | The space $M\_k(1)$ of modular forms of level $1$ has a unique basis $\left\{ f\_0,f\_1,\ldots,f\_d\right\}$ such that $a\_i(f\_j) = \delta\_{ij}$ for all $0\leq i,j \leq d$, and $f\_i \in \mathbb{Z}[[q]]$ for all $0 \leq i \leq d$. This is often called the *Miller basis* for $M\_k(1)$.
>
> Does there exist a basi... | https://mathoverflow.net/users/18394 | Is there a Miller basis for M_k(N)? | The space $M\_k(N)$ has a basis in $\mathbb{Z}[[q]]$ for any $N$ and $k$. This is a straightforward consequence of Eichler-Shimura theory and will be in any decent textbook on modular forms (e.g. Diamond + Shurman, Miyake, Lang).
| 3 | https://mathoverflow.net/users/2481 | 78048 | 47,078 |
https://mathoverflow.net/questions/77904 | 5 | For an algebraic number $\alpha$ one can define its "height" in many ways. Informally, you could use its minimal polynomial over $\mathbf{Q}$ and consider the maximum of the heights of its coefficients. Or consider all the valuations of $\alpha$, etc. In this context, the height is supposed to be some kind of measure o... | https://mathoverflow.net/users/18464 | Is there a reasonable definition of the height of a transcendental number | It does'nt quite answer your question, but maybe there is a resaonable definition of the height of a [period](http://en.wikipedia.org/wiki/Ring_of_periods). The ring of period is a countable over-ring of the field of algebraic numbers, sharing (at least conjecturally) many properties with them. This ring contains most ... | 2 | https://mathoverflow.net/users/13552 | 78056 | 47,084 |
https://mathoverflow.net/questions/78017 | 1 | Can anyone explain what is definition of maslov index in Heegaard Floer homology? I am puzzled> Thank you.,
| https://mathoverflow.net/users/18496 | Maslov Index in heegaard floer homology | Working in the Fukaya category (objects are Lagrangian submanifolds $L\_i$ of the symplectic 2n-manifold $M$ with some extra hypotheses), assume $2\cdot c\_1(M)=0$. Then we can trivialize $(\bigwedge^n\_\mathbb{C}TM)^2$ and choose a nowhere-vanishing section $\Omega\in\Gamma((\bigwedge^n\_\mathbb{C}TM)^2)$. This induce... | 1 | https://mathoverflow.net/users/12310 | 78062 | 47,088 |
https://mathoverflow.net/questions/78035 | 3 | To make my question precise, suppose you have a complex curve locally given by
$$f(x,y) =0 $$
and $f$ has singularity of type $\chi\_k$ at the origin. The codimension of
this singularity is $k$. Let $g$ be the contribution of the singularity to
the genus of that curve. For instance if it was a simple node ($A\_1$)... | https://mathoverflow.net/users/4463 | Is there an upper bound and a lower bound on the contribution to the genus, for a singularity of codimension k? | I assume you are talking about equisingular/topologically equivalent singularities (if you are talking about the analytical types, then the codimension is even higher). In that case, the relation can be computed from the embedded resolution of the singularity, as follows. This resolution consists in blowing up the poin... | 1 | https://mathoverflow.net/users/1939 | 78063 | 47,089 |
https://mathoverflow.net/questions/78065 | 10 | Good evening,
I have two questions concerning Cech cohomology of presheaves.
(1) Let $X$ be a topological space and $0\to\mathcal{F}\to\mathcal{G} \to \mathcal{H}\to 0$ a short exact sequence of sheaves of abelian groups on $X.$ Does there exist a long exact sequence for the Cech cohomology of these sheaves as the... | https://mathoverflow.net/users/11376 | Long exact sequence for Cech cohomology? | In a paracompact space, the answer to (1) is yes, and is done in Godement's book *Topologie algébrique et théorie de faisceaux*. Your last question in (2) is Theorem 5.10.2 there, and (2) is Theorem 5.10.1, and the question in youe edit is the Corollary to theorem 5.10.2. (You can replace the hypothesis on the space by... | 7 | https://mathoverflow.net/users/1409 | 78073 | 47,095 |
https://mathoverflow.net/questions/77499 | 0 | Consider finite-dimensional (for-simplicity) $\star$-algebras, that is, unital associative algebras over the complex numbers equipped with an antilinear antiautomorphism $\star$.
A state on a $\star$-algebra $A$ is a linear mapping $\psi: A \to \mathbb{C}$ satisfying
(i) $\psi(1) = 1$ [normalization]
(ii) $\psi(a... | https://mathoverflow.net/users/11146 | Mappings between states on *-algebras | Let me first stick to more conventional notation and, given a positive operator $f:C\rightarrow A$, denote by $f^\ast$ the map that you call $f^{-1}$. If $B=A\otimes C$ and $f$ is defined by $f(a\otimes c) = a\phi(c)$ with $\phi$ a state on $C$, then you get a special case of what is called a *conditional expectation,*... | 3 | https://mathoverflow.net/users/14756 | 78076 | 47,097 |
https://mathoverflow.net/questions/77763 | 28 | Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f\_T(z) = \sum\_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ordinary generating function for traces of exterior powers of $T$. Expressed another way,
$$ f\_T(z) = \mbox{Det}(I + zT) $... | https://mathoverflow.net/users/9068 | Can an operator have Exp(z) as its characteristic "polynomial"? | Here's a proof that $\exp(z)$ is not a characteristic function using the product expansion for the determinant, which is essentially equivalent to Lidskii's theorem stating that the trace of a trace class operator is the sum of its eigenvalues.
>
> If $T$ is a trace class operator on a Hilbert space $\mathcal{H}$ w... | 15 | https://mathoverflow.net/users/1004 | 78079 | 47,099 |
https://mathoverflow.net/questions/78055 | 3 | During my research I have recently stumbled upon the problem of finding the relative homotopy sets $\pi\_1(U\_n,U\_n/O\_n)$ and $\pi\_1(U\_{2n},U\_{2n}/USp\_{2n})$ for $n$ large enough to be in the stable regime where Bott periodicity applies.
The subgroup $U\_n/O\_n\subset U\_n$ contains all elements $g\in U\_n$ wi... | https://mathoverflow.net/users/18508 | The relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$ | There's a "famous" 24 term sequence of fibrations, involving the spaces of the real and complex K-theory spectra, coming from Bott periodicity. (But apparantly, not famous enough for google to cough up a good reference.)
Part of this sequence are homotopy fibration sequences of the form
$$ Sp/U \to U/O \xrightarrow{f} ... | 5 | https://mathoverflow.net/users/437 | 78088 | 47,104 |
https://mathoverflow.net/questions/78095 | 2 | Let $f:$ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ a $C^1$ diffeomorphism,
$x\in\Omega(f)$.$\space$
How do I prove that $\forall\space\epsilon\gt0$, $\exists\space$ $g:$$\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ with $d(f,g)\lt\epsilon$ $\space$
such that $x\in Per(g)$?
$\Omega(f)=$ { $y\in\mathbb{R}\mid\space\forall\sp... | https://mathoverflow.net/users/18231 | Closing lemma on the Interval | Since your $f$ is a diffeo, it is monotone, and a wandering point of a monotone map on $\mathbb{R}$ is already either a fixed point or a point of period 2 (the latter can only happen if $f$ is decreasing).
| 2 | https://mathoverflow.net/users/6101 | 78100 | 47,110 |
https://mathoverflow.net/questions/78106 | 16 | This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? What are the existence and uniqueness conditions for such solution?
| https://mathoverflow.net/users/18526 | Solving a quadratic matrix equation | This relationship is called congruence, and for symmetric (or hermitian) matrices $A, B$ such an $X$ exists if and only if the matrices have the same inertia (the same number of positive, negative, zero eigenvalues). Fro general matrices congruence is much less well known, but it *is* quite well understood, see:
<htt... | 14 | https://mathoverflow.net/users/11142 | 78116 | 47,115 |
https://mathoverflow.net/questions/78115 | 4 | I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.
| https://mathoverflow.net/users/1937 | References about pseudoeffective cone | I'd say Lazarsfeld's book "[Positivity in algebraic geometry I,II](https://books.google.com/books/about/Positivity_in_Algebraic_Geometry_I.html?id=T87ftUcU_hEC "zbMATH review at https://zbmath.org/1066.14021")" is the standard reference these days. In particular, Volume I has a lot of explicit examples. I also recommen... | 13 | https://mathoverflow.net/users/3996 | 78117 | 47,116 |
https://mathoverflow.net/questions/78038 | 5 | Given a k-regular graph $G$ (every vertex is of degree k), one defines its Laplace operator as
$L(G)=D-A=kI-A$, where $I$ is identity matrix and $A$ adjacency matrix of $G$.
Let $\lambda\_{1}\leq \ldots \leq \lambda\_{n}$ be eigenvalue of $L(G)$,
are there any results on lower bound on $\lambda\_{n}$. Of course, here ... | https://mathoverflow.net/users/18502 | Bounds on maximal eigenvalue of a k-regular graph | Even though the question has been answered, I feel I should flesh out my comment into an answer. Apologies for the redundancy.
First, here's a simple proof of this special case of the Hoffman bound. Suppose $X$ is an independent set of vertices of $G$ (some call this a coclique) of cardinality $\alpha$. Let $\mathbf{... | 4 | https://mathoverflow.net/users/14913 | 78131 | 47,120 |
https://mathoverflow.net/questions/78132 | 11 | I would like to know what kind of analogs of Kodaira vanishing theorem are valid for singular varieties. For example, is the following true: let $X$ be a projective Gorenstein variety and let $\omega\_X$ be its canonical bundle. Is it true that $H^i(L\otimes \omega\_X)=0$
for $i>0$ for an ample line bundle $L$?
| https://mathoverflow.net/users/3891 | Is there an analog of Kodaira vanishing for singular varieties | No. The following counterexample is due to Sommese:
Let $Y$ be the projective bundle $\pi:\mathbb{P}(O\oplus O(1)^{\oplus 3})\to \mathbb{P}^1$. Let $M$ be the tautological bundle on $Y$ and take a general member $X\in|M\otimes \pi^\*O(-1)^{\oplus 4})|$. Then $X$ is a normal, projective, Gorenstein 3-fold. If $L$ is t... | 17 | https://mathoverflow.net/users/3996 | 78135 | 47,122 |
https://mathoverflow.net/questions/78129 | 4 | I have casually almost (i.e. up to details that shoud work) proved the following discrete version of Brouwer's fixed point theorem. I should have obtained this result as a corollary of quite complicated things and I do not understand if the result is trivial and can be easily proved directly or it deserves to be stress... | https://mathoverflow.net/users/13809 | A Brouwer fixed point theorem on finite sets | I believe the following is a counter-example:
$f: \lbrace -1,0,1\rbrace^2 \to \lbrace -1,0,1\rbrace^2$
$\forall x:$
$ f(-1,x) = (1,x)$
$f(0,x)=(1,x)$
$f(1,x)=(0,x)$
| 6 | https://mathoverflow.net/users/16447 | 78137 | 47,123 |
https://mathoverflow.net/questions/77353 | 5 | Let $\Delta = \sigma + 4 m$ be the fundamental discriminant of a quadratic field, where $\sigma \in \{ 0, 1 \}$. The binary quadratic form $Q(x, y) = A x^2 + B x y + C y^2$ of discriminant $\Delta$ belongs to the identity class (principal) of the narrow class group of forms, under substitution by $(x,y) \mapsto M (x', ... | https://mathoverflow.net/users/17053 | What is the identity class of the set of equivalence classes of binary cubic forms of discriminant $D$ ? | I'm not sure that I will be answering your question, so let me first recall the background. The equivalence classes of primitive binary cubic forms with discriminant $\Delta$ correspond to $SL\_2(\mathbb Z)$-equivalence classes of triply symmetric Bhargava cubes. There is a natural homomorphism from this group to the s... | 3 | https://mathoverflow.net/users/3503 | 78139 | 47,125 |
https://mathoverflow.net/questions/78147 | -1 | I've casually proved, as application of some ideas that I am developing, a result that might be of interest in itself. I am completely new in this field and then I would like to ask your help to understand: 1) might it be of interest? 2) Is it trivial, in the sense that it can be proved directly? 3) is it well-known?
... | https://mathoverflow.net/users/13809 | Walks that cannot hit the boundary | In fact, you can prove that $p^+=p$ for all $p$.
Actually, consider the shortest paths from $(-n,-n)$ to $(n,n)$ and from $(-n,n)$ to $(n,-n)$ passing through $p$. Taking `pluses' of them, you should also obtain the paths of the same lengths connecting the same points. Now from the first path you obtain that the sum ... | 5 | https://mathoverflow.net/users/17581 | 78151 | 47,132 |
https://mathoverflow.net/questions/78159 | 6 | The following is a weaker version of what is called splitting principle in
[Appendix C, page 12](http://www.ma.huji.ac.il/~karshon/monograph), see also for a lighter version [Brions Eq cohom and eq intersection theory, page 6](http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html):
Let $G$ be a compact (complex)... | https://mathoverflow.net/users/18539 | Splitting principle in equivariant cohomology | The embedding of the unitary group $U\_n$ into $GL\_n(\mathbb C)$ is a homotopy equivalence; this is easily seen to imply that $H^\*\_{U\_n}(X)$ is isomorphic to $H^\*\_{GL\_n}(X)$. So the result for compact groups implies that for $GL\_n$.
The same idea works for any reductive complex algebraic group $G$, since the ... | 5 | https://mathoverflow.net/users/4790 | 78161 | 47,135 |
https://mathoverflow.net/questions/78119 | 7 | Fejer's theorem says that for any continuous function $f \colon S^1 \to \mathbb C$ with Fourier coefficients $a=(a\_n)\_{n \in \mathbb Z}$ the sequence
$$\sigma\_n(a) := \frac1n \sum\_{k=1}^n \sum\_{l=-k}^k a\_l \exp(2\pi i \cdot l\phi)$$
convergences uniformly to $f$. Moreover, by the Riemann-Lebesgue Lemma, the s... | https://mathoverflow.net/users/8176 | Fejer's theorem and convergence of Fourier series in measure | There is no continuous linear operator from $c\_0$ to $M(S^1)$ that maps the unit vector basis to the characters. In fact, any continuous linear operator from $c\_0$ to $M(S^1)$ maps the unit vector basis to a sequence which converges to zero at a good rate. To see this, note that by Maurey-Nikishin, the operator facto... | 4 | https://mathoverflow.net/users/2554 | 78162 | 47,136 |
https://mathoverflow.net/questions/78172 | 1 | Nowadays, we know that there exist Banach spaces without unconditional basic sequences. Do we know if something a bit milder holds? Namely, is that true each non-reflexive Banach space contains a normalised basic sequence $(x\_n)\_{n=1}^\infty$ such that for each $(t\_n)\_{n=1}^\infty\in c\_0$ the series
$$\sum\_{n=1... | https://mathoverflow.net/users/18542 | Basic sequences | The Banach spaces that admit such a sequence are the Banach spaces that contain a subspace isomorphic to $c\_0$. Look at, e.g., the beginning part of the book of Albiac-Kalton.
| 2 | https://mathoverflow.net/users/2554 | 78174 | 47,143 |
https://mathoverflow.net/questions/78175 | 13 | The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for example, S. MacLane, Categories for Working Mathematicians). To every topological space this left adjoint associates a Haus... | https://mathoverflow.net/users/18543 | Largest Hausdorff quotient | Consider the equivalence relation $\sim$ on your space $X$ such that $x\sim y$ iff $x$ and $y$ have the same image under all surjective continuous maps $f:X\to Y$ with codomain $Y$ a Hausdorff space. Put on the set $X/\sim$ the least topology which makes all those maps continuous, and you have the space you want. I dou... | 20 | https://mathoverflow.net/users/1409 | 78182 | 47,147 |
https://mathoverflow.net/questions/77748 | 1 | Consider a class of real symmetric random matrices,$M\_{n\times n}=(X\_{i,j})\_{n\times n},$ whose off-diagonal elements follow an exchangeable distribution, and the diagonal elements follow another exchangeable distribution and the diagonal part and off-diagonal part are independent. My question is what's the eigenval... | https://mathoverflow.net/users/18420 | Eigenvalue density of some random matrices? | If you don't require anything else besides exchangeable, then I guess not much can be said. Since exchangeability includes the trivial case of completely coupled random variables. Say consider the following matrix: the diagonal entries are all the same variable $X$, and off diagonal ones the same variable $Y$, independ... | 2 | https://mathoverflow.net/users/4923 | 78189 | 47,149 |
https://mathoverflow.net/questions/73063 | 6 | Recall that a *Poisson algebra* is a commutative algebra $A$ along with a bracket $\lbrace,\rbrace: A^{\otimes 2} \to A$ which is a Lie bracket and which is also a derivation in each variable. The *Poisson center* of $A$ is the subalgebra of those $f\in A$ such that $\lbrace f,\rbrace : A \to A$ is the $0$ derivation. ... | https://mathoverflow.net/users/78 | How can I see the "missing" Poisson center when the rank of the Poisson structure drops? | I do not have a definite answer but rather some reflections I've being doing myself and with a colleague I'll mention later, recently, on the subject.
Sure Poisson cohomology can help you in detecting the"missing" leaves, my favourite example being the triple of bivectors $\partial\_x\wedge\partial\_y$ (trivial 1-Poiss... | 2 | https://mathoverflow.net/users/6032 | 78197 | 47,153 |
https://mathoverflow.net/questions/78194 | 40 | If $\mathbf{P}^1$ is replaced by the affine line $\mathbf{A}^1$, this becomes the cancellation problem, and we have a pair of famous Danielewski surfaces ($xy=1-z^2$ and $x^2y=1-z^2$) as a counterexample (though I'm still seeking how to prove that..). I suppose in my case this counterexample might no longer work.
Al... | https://mathoverflow.net/users/10333 | $V$, $W$ are varieties. Does $V\times \mathbf{P}^1=W\times \mathbf{P}^1$ imply $V=W$? | This problem was studied by Fujita in his paper ["Cancellation problem of complete varieties", Inventiones Mathematicae 64 (1981).](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356556735_0064&DMDID=dmdlog13)
He showed that the obstruction to cancellation is caused by the Picard schemes, proving the followin... | 80 | https://mathoverflow.net/users/7460 | 78198 | 47,154 |
https://mathoverflow.net/questions/78196 | 0 | Let X be a Go space. If G is open in X, why is every convex component of G open?
( It is well known that any non-void subset G of X can be uniquely represented as a union
of its maximal convex subsets, which are called convex components of G. )
| https://mathoverflow.net/users/18465 | On generalized ordered spaces | Lemma: if $K$ and $L$ are (order) convex in a linearly ordered set $X$, and $x$ is in $K \cap L$, then $K \cup L$ is convex as well.
Proof: suppose $a < b$ are in $K \cup L$ and $c$ lies in $(a,b)$. Consider 2 cases:
1) $c \le x$: then $a$ is in one of the convex sets, say $K$, and so is $x$, but then
$x \in (a, x... | 2 | https://mathoverflow.net/users/2060 | 78207 | 47,157 |
https://mathoverflow.net/questions/78165 | 10 | A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each point on the ellipse there is exactly one such a maximizing triangle with vertex on that point, which can also be viewed... | https://mathoverflow.net/users/6101 | Convex curves with many inscribed triangles maximizing perimeter | N.B. This is an edit of my original post, confirming the guess that I made originally.
The answer is *no*, i.e., such curves are not forced to be ellipses.
Here is a sketch of the argument. (The details will take a while to type in, and that will have to wait. Also, I'm not sure that there will be that many people ... | 10 | https://mathoverflow.net/users/13972 | 78213 | 47,159 |
https://mathoverflow.net/questions/78185 | 15 | I am writing two separate paper that are closely related. When I try to submit to arXiv, is it possible for each paper to refer to the other paper with an arXiv link, rather then putting a newer version of one of them just to replace the arXiv link in the reference? Since I heard from others that too many frequent upda... | https://mathoverflow.net/users/18550 | Double Referencing in arXiv | I asked the arxiv administrators about this last year, and received the following reply.
---
Dear Scott,
Thank you for your feedback. At this time we have no plans to change
the way the submission ID and the final arXiv ID are created. We
developed the new submission system to be more flexible for our users
... | 18 | https://mathoverflow.net/users/3 | 78217 | 47,161 |
https://mathoverflow.net/questions/78220 | 3 | Is it true that every closed operator on a separable Hilbert H space only has countably many eigenvalues?
Or put the other way around, if I want to ensure that a (not necessarily bounded) linear operator on a separable Hilbert space only has countably many eigenvalues, is closedness (or better said, closability) a su... | https://mathoverflow.net/users/16702 | Countability of eigenvalues of a linear operator | Let $T:\ell^2\rightarrow\ell^2$ be the backwards shift operator, $T(a\_n) = (a\_2,a\_3,\cdots)$. This is a contraction. For any $\lambda\in\mathbb C$, consider the sequence given by $a\_n = \lambda^n$. Thus $(a\_{n+1}) = (\lambda^2,\lambda^3,\cdots) = \lambda(\lambda,\lambda^2,\cdots)$ and so, if $(a\_n)\in\ell^2$, the... | 13 | https://mathoverflow.net/users/406 | 78224 | 47,164 |
https://mathoverflow.net/questions/78060 | 28 | We all know mathematics is life, this question is for Mankind. It's mathoverflow here when some parts of the world we have mathunderflow! I think we can do something through ideas. A similar question "Good ways to engage in mathematics outreach" has been featured but this is a different question all together.
In this... | https://mathoverflow.net/users/1997 | Means of Promoting Mathematics in Young Countries! | Dear Ongaro Nyang'
Although Israel is no longer so young, it was young, (even
younger than most other countries,) not so long ago, and
it is a small and rather isolated place, with some difficulties.
So some lessons from Israeli mathematics, especially in its early days
may be relevant.
>
> A) Immigration
>
... | 19 | https://mathoverflow.net/users/1532 | 78231 | 47,168 |
https://mathoverflow.net/questions/78236 | 3 | I would like to know what are the group cohomology classes $H^d[Z\_n, Z\_2]$, $H^d[U(1), Z\_2]$, $H^d[SO(n), Z\_2]$, $H^d[SU(n), Z\_2]$, etc. Thanks!
(Here the group cohomology $H^d[G, M]$ for a group $G$ is the topological cohomology of the classifying space $BG$, $H\_{top}^d[BG, M]=H^d[G, M]$.)
| https://mathoverflow.net/users/17787 | Group cohomology with $Z_2$ coefficient | For the latter three, here is the integer-coefficients (apply Kunneth formula to get your mod-2 coefficients:
[Group cohomology of compact Lie group with integer coeffient](https://mathoverflow.net/questions/75389/group-cohomology-of-compact-lie-group-with-integer-coeffient)
As for the first: $H^i(\mathbb{Z}\_n)$ ... | 3 | https://mathoverflow.net/users/12310 | 78237 | 47,171 |
https://mathoverflow.net/questions/78240 | 15 | While browsing through some papers, I came across some literature discussing the Arthur-Selberg trace formula. At a conceptual level I think I understand what it is doing, but when I get down to the technical details I start to get a bit lost. Part of the problem is that James Arthur's papers are all written using adel... | https://mathoverflow.net/users/4642 | What problem do the adeles solve? | When working on a rational problem (over $\mathbb{Q}$), you can't do much analysis - so you lack quite a few tools.
The obvious solution is then to pass to the completion, where you'll be able to do analysis ; so most people go to $\mathbb{R}$. But that isn't that natural : $\mathbb{Q}$ has several completions in fac... | 23 | https://mathoverflow.net/users/12664 | 78253 | 47,180 |
https://mathoverflow.net/questions/78252 | 4 | Let $f:Y\rightarrow X$ be an etale morphism, where $X$ and $Y$ are smooth projective
varieties. Let $V$ be a vector bundle over $Y$. Since $f$ is flat, $V$ is flat over $X$.
Is it true that $f\_\*V$ is flat $\mathcal{O}\_X$-module?
| https://mathoverflow.net/users/17420 | etale morphism and direct image | Since $Y$ and $X$ are proper over the base field, the morphism $f$ is finite (by Zariski's main theorem). Hence by the semi-continuity theorem (see Hartshorne p. 288, Cor. 12.9), $f\_\*V$ is locally free, because $f$ is finite and flat. So the answer is yes.
| 5 | https://mathoverflow.net/users/17308 | 78254 | 47,181 |
https://mathoverflow.net/questions/78255 | 6 | Let $S$ be the Siegel-half plane of dimension $n$, i.e. the set of complex $n \times n$ matrices $Z$ which are symmetric and whose imaginary part is positive-definite. In dimension 1 we can identify $S$ with the Poincaré half-plane.
In dimension 1 there is an action of the symplectic group $Sp\_2(\mathbb R)$ on $S$, ... | https://mathoverflow.net/users/4054 | Finding the action of the symplectic group on the Siegel-half plane | The Siegel half-plane is just one example of a very general construction: if one lets $G$ be a semisimple Lie group, and $K$ a maximal compact subgroup, then $G$ acts on the quotient $G / K$, and the quotient is known as a "symmetric space". The Siegel upper half-plane is just a symmetric space for $Sp\_{2n}(\mathbb{R}... | 7 | https://mathoverflow.net/users/2481 | 78257 | 47,183 |
https://mathoverflow.net/questions/23143 | 29 | Tom Leinster has a note [here](http://www.maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf) about how you can realize L^1[0,1] as the initial object of a certain category. You should really read his note because it is only 2.5 pages and is much more charming than what I am going to write below as background, but if you don't ... | https://mathoverflow.net/users/1106 | What theorem constructs an initial object for this category? (Formerly "Integrability by abstract nonsense") | I hope this answers the question a bit more explicitely as the other one. The general theorem which applies here is the following:
>
> Theorem: Let $C$ be a category which as an initial object and colimits of $\omega$-chains. Then for every functor $F : C \to C$ which preserves these colimits, there exists an initi... | 21 | https://mathoverflow.net/users/2841 | 78262 | 47,185 |
https://mathoverflow.net/questions/78260 | 14 | At the end of Section 14.1 in Pressley, Segal "Loop Groups" there is the remark that
the partition function is a modular function in the sense that the Dedekind $\eta$ function is a modular form. I am interested about the following remark:
>
> It follows from the Kac character formula that the characters of all po... | https://mathoverflow.net/users/10718 | What is known about the connection of positive energy representations of loop groups and modular forms | That the characters of representations have modular properties is the content of Kac-Peterson [1]. As for an "explanation" of this, there are several but I'm afraid all of the ones I know involve in one way or the other physics. In his seminal paper [2] Zhu proves modularity for Vertex Operator Algebras with certain co... | 15 | https://mathoverflow.net/users/17980 | 78263 | 47,186 |
https://mathoverflow.net/questions/78160 | 7 | Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra, and $M(\Lambda)$ the Verma module with dominant integral highest weight $\Lambda$. The singular vectors in $M(\Lambda)$ are well known (they are something like $f\_i^{\Lambda(h\_i^\vee)+1} v\_\Lambda$). A similar fact is true for the affine Lie algebra $\wi... | https://mathoverflow.net/users/18540 | Singular vectors in Verma modules. | This was supposed to be a comment but grew too much. Some progress using the lines of work of the article you cite is given in [1] where the author gives explicit formulas (he gives an algorithm), he uses rational powers of nilpotent elements and he sketches that his algorithm works in the affine case as well. Also Fei... | 3 | https://mathoverflow.net/users/17980 | 78264 | 47,187 |
https://mathoverflow.net/questions/77866 | 2 | Let $X$ be a locally compact separable & metrizable space, and $M^+(X)$ its space of positive measures (i.e. positive linear forms on the space of continuous functions on $X$, continuous on each space of continuous functions with a support included in a given compact $K$ of $X$). It is not difficult to show that if we ... | https://mathoverflow.net/users/3333 | Separability of sets of positive measures | This was also posted on [Math.SE](https://math.stackexchange.com/questions/71150/separability-of-the-set-of-positive-measures). I'm reposting my answer from there.
I think I'm with you; the exercise in Dieudonné seems to be in error.
I certainly agree that $M^+(X)$ can be separable, for $X$ non-compact. Take for in... | 1 | https://mathoverflow.net/users/4832 | 78267 | 47,189 |
https://mathoverflow.net/questions/78228 | 6 | Let $X\subset \mathbb{P}^5$ a smooth pfaffian smooth cubic fourfold hypersurface. It is easy to see that such a hypersurface must contain a quartic scroll surface. I wonder about the inverse question. If a cubic fourfold $X$ contains a quartic scroll, is it a pfaffian?
| https://mathoverflow.net/users/4096 | Are cubic four-folds containing a quartic scroll pfaffians? | Part (a) of Proposition 9.2 in Beauville's "Determinantal Hypersurfaces" paper (Michigan Mathematical Journal 48, 2000) says that a cubic fourfold is linear Pfaffian precisely when it contains a quintic del Pezzo surface. One path to settling your question is to determine whether every cubic fourfold $X$ containing a q... | 3 | https://mathoverflow.net/users/5496 | 78271 | 47,191 |
https://mathoverflow.net/questions/78275 | 8 | For a higher genus Riemann surface $\Sigma$, is it true that every nontrivial (holomorphic) automorphism is of nontrivial mapping class, i.e., not isotopic to the identity?
| https://mathoverflow.net/users/2555 | Automorphisms of Riemann surface and mapping class | A much more general fact is true: *any* isometry of *any* closed negatively curved Riemannian manifold is not homotopic to the identity. There are many proofs of this; one (perhaps not the most natural) is as follows. Hartman proved that if two harmonic maps $f\_0,f\_1\colon M\to N$ are homotopic, where $M$ is compact ... | 12 | https://mathoverflow.net/users/250 | 78289 | 47,199 |
https://mathoverflow.net/questions/78287 | 11 | If $G$ is a finite group with periodic cohomology then the Tate cohomology ring can be easily computed to be the localization $\hat{H}^\ast(G,\mathbb{Z}) = H^\ast(G,\mathbb{Z})\_{(z)}$ where $z$ is a unit of minimal positive degree. Examples are
* Cyclic group:
$\hat{H}^\ast(C\_n,\mathbb{Z}) = \mathbb{Z}[z,z^{-1}... | https://mathoverflow.net/users/18571 | Examples of Tate cohomology rings | Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}\_p^2$. The title is *Integral Duality and the Structure of Tate Cohomology Rings*.
Also, there is an unpublished/unfinished paper of... | 6 | https://mathoverflow.net/users/12310 | 78290 | 47,200 |
https://mathoverflow.net/questions/78300 | 3 | The Peano Axioms (partially) formalize our intuitive notion of arithmetic. Partially because they also describe the behaviour of nonstandard models and there are some theorems that they can not prove that might seem, at first sight, to be within their domain e.g. Goodstein's Theorem and some of Harvey Friedman's combin... | https://mathoverflow.net/users/18578 | Extension of the Peano Axioms? | There is no hope for a *first-order* theory to eliminate non-standard models. If a first-order theory over a countable language has an infinite model then it has models of all infinite cardinalities ([Löwenheim-Skolem Theorem](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem)). In the case of a theory... | 7 | https://mathoverflow.net/users/2000 | 78301 | 47,205 |
https://mathoverflow.net/questions/78247 | 11 | In the one-dimensional case the Langlands program is equivalent to the class field theory and the two-dimensional case implies the Taniyama-Shimura conjecture.
I would like to know: are there any other important consequences of the Langlands program?
| https://mathoverflow.net/users/18503 | Consequences of the Langlands program | There are many, many consequences of the general Langlands program (which I'll interpret to mean both functoriality for automorphic forms and reciprocity between Galois representations and automorphic forms). Some of these are:
* The Selberg $1/4$ conjecture.
* The Ramanujan conjecture for cuspforms on $GL\_n$ over a... | 26 | https://mathoverflow.net/users/2874 | 78304 | 47,207 |
https://mathoverflow.net/questions/78250 | 10 | Say we have some equations $f\_1(x)=0, \ldots f\_k(x)=0$ defining a variety $X$ in ${\mathbb C}^n$ (not necessarily a minimal number of generators, and not necessarily of minimal degree), and suppose we want to know the ideal $\bar{I}$ of its closure $\overline{X}$ in ${\mathbb P}^n$. A naive question:
> If all the ... | https://mathoverflow.net/users/84526 | Can taking the projective closure of an affine variety increase the degrees of its ideal generators? | Consider $R = \Bbbk[x\_1, \ldots, x\_5]$ and $I = (x\_1x\_2^2-x\_3^2, x\_1x\_4^2-x\_5^2)$. When we homogenize w.r.t $z$, we get $(x\_1x\_2^2-zx\_3^2, x\_1x\_4^2-zx\_5^2)$ whose saturation w.r.t $z$ contains the binomial $x\_3^2x\_4^2-x\_2^2x\_5^2$ of degree $4$. The original generators are of degree at most $3$.
Loo... | 8 | https://mathoverflow.net/users/14895 | 78308 | 47,210 |
https://mathoverflow.net/questions/78284 | 2 | I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom
| https://mathoverflow.net/users/14574 | Non-split groups | Besides the basic definitions and examples, you will find a concise description of the vocabulary needed to talk about linear algebraic groups over fields that are not algebraically closed in T. A. Springer's article titled *Reductive Groups*, which appears in Part I of *Automorphic Forms, Representaions, and L-Functio... | 3 | https://mathoverflow.net/users/9672 | 78309 | 47,211 |
https://mathoverflow.net/questions/78312 | 1 | Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F\_v$ of $F$ and $G\_v = GL(2,F\_v)$. Now the eigenvalues of $\gamma$ may or may not lie in $F\_v$. What is the centralizer
$$ C\_{G\_v}(\gamma... | https://mathoverflow.net/users/10400 | Centralizer of elliptic elements in $GL(2)$ | The centralizer of $\gamma$ is a torus of the form $E^\times$, where $E/F$ is a quadratic field extension. Assume moreover that $\gamma$ is regular (so that $E/F$ is separable). The centralizer of $\gamma$ in $G\_v$ is $(E\otimes\_F F\_v )^\times$. The algebra $E\otimes F\_v$ is either a field (in that case $\gamma$ is... | 3 | https://mathoverflow.net/users/4767 | 78316 | 47,213 |
https://mathoverflow.net/questions/78313 | 0 | Hi,
this is a bit vague, but feel free in your answers.
If I have a scheme $X$ and a closed point $x$ on it, then how is the first infinitesimal neighborhood of the point defined, and what is the philosophy behind it? What is it useful for, when does it appear?
In particular, I would be interested in the case whe... | https://mathoverflow.net/users/18580 | First order infinitesimal neighborhood of a point | Let $X$ be a scheme and $x$ a closed point. Let us call $\mathcal{I\_x}$ the sheaf of ideals in $\mathcal{O}\_X$ that defines $x$ Notice that the canonical embedding $x \hookrightarrow X$ is given by the sheaf map $\mathcal{O}\_X \to \mathcal{O}\_X/\mathcal{I\_x}$. Now consider the ideal $\mathcal{I}\_x^2$ and the clos... | 7 | https://mathoverflow.net/users/6348 | 78320 | 47,215 |
https://mathoverflow.net/questions/78314 | 5 | In Adams, J.F. *Infinite Loop Spaces* Princ. Univ. Press. page 9 he states Alexander duality theorem
**Theorem:[Alexander Duality]** $$ H^r(X,G)=H\_{n-r+1}(S^n-X,G)$$
for finite CW-complexes with a "nice embedding". That is to say, $S^n-X$ has a CW-complex $Y$ as deformation retract and $X$ is a deformation retrac... | https://mathoverflow.net/users/12204 | Alexander duality theorem for CW-complexes and stable homotopy theory | 1) The answer is yes, at least up to homotopy. This can be found in Wall's paper:
Wall, C. T. C., Classification problems in differential topology---IV. Thickenings. Topology, 1966, 5, 73–94.
Wall argues this in a cell-by-cell way, making use of transversality.
2) Alternatively, if X is a finite simplicial comple... | 7 | https://mathoverflow.net/users/8032 | 78327 | 47,219 |
https://mathoverflow.net/questions/78328 | 2 | Berkeley's collection of past qualifying exam questions contains the following:
''What are possible extensions of degree $3$ of $\mathbb{Q}\_2$?''
I'm trying to figure out what the general approach is to attack a question like this. In this particular case, we know that $\mathbb{Q}\_2^\times\simeq \mathbb{Z}\times ... | https://mathoverflow.net/users/18586 | Enumerating non-abelian extensions of $\mathbb{Q}_p$? | This is standard stuff. Here is (in French) the solution as an exercise, copy-pasted from the final exam of a course I gave on local fields.
Soit $K$ une extension totalement ramifiée de degré $n$ de $Q\_p$ et $\pi\_K$ une uniformisante de $K$. On suppose pour l'instant que $p \nmid n$.
1. Montrer que si $w \in Q\_... | 6 | https://mathoverflow.net/users/5743 | 78329 | 47,220 |
https://mathoverflow.net/questions/78330 | 10 | $\DeclareMathOperator\Tr{Tr}$Let $A\_i$ with $i=1,\dotsc,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $\Tr(p)=1$. Let $0< a\_i<1$ and $\sum\_{i=1}^N a\_i = 1$.
Claim:
$$\Tr( A\_1 p^{a\_1} A\_2 p^{a\_2} \dotsm A\_N p^{a\_N} ) \leq \lVert A\_1\rVert \lVert A\_2\... | https://mathoverflow.net/users/18587 | Bounding the trace of a matrix product by the operator norms; generalized Hölder inequality? | The claim is true. More generally the Hölder inequality holds for the Schatten $p$-norms. The statement is [here without proof](https://en.wikipedia.org/wiki/Schatten_class_operator) in Wikipedia, and by induction it implies that for matrices $X\_1,\dotsc, X\_K$ and $p\_1,\dotsc,p\_K \in [1,\infty]$ with $\sum\_i 1/p\_... | 12 | https://mathoverflow.net/users/10265 | 78331 | 47,221 |
https://mathoverflow.net/questions/78018 | 10 | I just had a look to the article [*The set theoretical multiverse*](http://arxiv.org/pdf/1108.4223v1) by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, but I was fascinated by the possible philosophical perspective of being compelled (... | https://mathoverflow.net/users/4721 | Set-theoretical multiverse and foundations | (It happens that I will be giving [a talk this week](http://boolesrings.org/hamkins/multiverse-harvard-2011/) on this topic at the [Exploring the Frontiers of Incompleteness](http://logic.harvard.edu/efi.php) series at Harvard, which is focussing on the question of determinism in set theory. The materials for all the t... | 10 | https://mathoverflow.net/users/1946 | 78343 | 47,224 |
https://mathoverflow.net/questions/78340 | 14 | Consider the commutative diagram below with exact rows (from the long exact sequence of homotopy groups) and $f\_1,f\_2,f\_4,f\_5$ bijective ($f\_1,f\_2$ homomorphisms). Does it follow that $f\_3$ is also bijective?
\begin{align}
\matrix{
\pi\_1(A)&\to&\pi\_1(X)&\to&\pi\_1(X,A)&\to&\pi\_0(A)&\to&\pi\_0(X)
\cr \downar... | https://mathoverflow.net/users/18508 | Five-lemma for the end of long exact sequences of homotopy groups | This is covered in Exercise 9 in Section 4.1 of [Allen Hatcher's book](http://www.math.cornell.edu/~hatcher/AT/ATpage.html) (page 358). It turns out that if (for all choices of base-points) $f\_1,f\_2,f\_4,$ and $f\_5$ are all bijections then $f\_3$ is a bijection. Note that you can also extend one more term to the rig... | 17 | https://mathoverflow.net/users/11540 | 78346 | 47,225 |
https://mathoverflow.net/questions/78347 | 2 | Let $H$ be an infinite dimensional separable complex Hilbert space with Lie group action (I am mostly interested in the case $S^1$). Let $\text{Gl}\_{G}(H)$ be the space of invertible, bounded and equivariant linear maps (from $H$ to $H$).
Now, in the non-equivariant case, Kuiper's theorem states that $\text{Gl}(H)$ ... | https://mathoverflow.net/users/3816 | Theorem of Kuiper for Hilbert spaces with group action | Assume that $G$ acts on $H$ through a unitary irreducible representation. Then by Schur's lemma, $GL\_G(H)$ is $\mathbb{C}^\times$, which is of course not contractible.
For $G=S^1$: consider the left regular representation on $L^2(S^1)$. Then $GL\_G(H)$ is the multiplicative group of bounded sequences with values in ... | 7 | https://mathoverflow.net/users/14497 | 78351 | 47,227 |
https://mathoverflow.net/questions/78332 | 1 | Hello,
I am reading Rosenstein's "Linear Orderings" and I am not sure if I am missing something, or if there is an error.
He gives the definition of a $\beta$-limit ordinal inductively, as follows (his exact wording is given):
1. $L\_0 = $ all ordinals
2. $\alpha\in L\_{\beta+1}$ if and only if $\alpha=\lim \{\al... | https://mathoverflow.net/users/15735 | Definition of $\beta$-limit ordinals | Not only the remark right after the definition, but also Exercise 5.2 ($\alpha$ is a $\beta$-limit ordinal iff the last exponent in the Cantor normal form of $\alpha$ is at least $\beta$) suggest that Rosenstein meant to allow any cofinality, as you suggested. This definition seems very natural, but I do not know if it... | 4 | https://mathoverflow.net/users/14915 | 78358 | 47,229 |
https://mathoverflow.net/questions/78350 | 1 | Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection of the tautological line bundle $(\mathcal O\_X(1))^n$.
| https://mathoverflow.net/users/1937 | Top self-intersection of the tautological line bundle | Let $\xi \in A^1(X) $ be the class of $\mathcal{O}\_X(1)$.
Since $\dim Y=1$, by [Hartshorne, Algebraic Geometry, p. 429] we have the equality $$\xi^n=\pi^\*c\_1(\mathcal{E}) \cdot \xi^{n-1}.$$
But $\mathcal{O}\_X(1)$ is a relative hyperplane, so $\pi^\*(\textrm{point}) \cdot \xi^{n-1}=1$. Then $$\xi^n= \deg (\mathc... | 4 | https://mathoverflow.net/users/7460 | 78359 | 47,230 |
https://mathoverflow.net/questions/78341 | 10 | I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.
However, I cannot remember where I read this, and nor can I find another place which says it. I was therefore wondering if some... | https://mathoverflow.net/users/6503 | residually finite-by-$\mathbb{Z}$ groups are residually finite | The modified question has a positive answer if $N$ is finitely generated.
Consider an extension $1 \to N \to G \to \mathbb Z \to 1$ and take a lift $u \in G$ of the generator of $\mathbb Z$. If $N$ is finitely generated and $H' \subset N$ is a subgroup of finite index, then the intersection of all subgroups of index ... | 10 | https://mathoverflow.net/users/8176 | 78360 | 47,231 |
https://mathoverflow.net/questions/78336 | 1 | Let $X$ be a countable set and $\mathcal M(X)$ be the set of **finitely additive** probability measures on $X$. If $\mu\in\mathcal M(X)$, I define the entropy of $\mu$ to be
$$
E(\mu)=\sup\left\{\left.-\sum\mu(A\_i)\log(\mu(A\_i))\\ \right|\\ X=\bigcup A\_i, A\_i \text{ pairwise disjoint}\right\}
$$
If $\mu$ is a D... | https://mathoverflow.net/users/13809 | existence of finitely additive measures with zero entropy | Posted above as a comment, but re-posted here as an answer following suggestion of @Benoit Kloeckner so original q does not appear unanswered.
The first question is equivalent to the existence of a non-principal ultrafilter: a measure has entropy 0 if and only if every set has measure 0 or 1. This question was asked... | 2 | https://mathoverflow.net/users/11054 | 78376 | 47,241 |
https://mathoverflow.net/questions/78270 | 21 | This is related to Victor Protsak's approach to [this question](https://mathoverflow.net/questions/26358/can-we-color-z-with-n-colors-such-that-a-2a-na-all-have-different-colors).
>
> Suppose that $p\gt 11$ is a prime of
> the form $5n+1$. Can we prove that
> $1^5,2^5,\dots,n^5$ *cannot* be
> pairwise different... | https://mathoverflow.net/users/6085 | Fifth powers modulo a prime | Following Darij Grinberg's comments I obtained
**Theorem.** For any integer $k>2$ there are only finitely many primes of the form $p=kn+1$ such that $1^k,2^k,\dots,n^k$ are distinct modulo $p$.
**Proof.** Assume that $k,n>2$ and $p=kn+1$ is a prime such that $1^k,2^k,\dots,n^k$ are distinct modulo $p$. Then the lis... | 14 | https://mathoverflow.net/users/11919 | 78377 | 47,242 |
https://mathoverflow.net/questions/78367 | 1 | Let $B=\int\_{Y}^{\oplus}B\_yd\_y$ be the direct integral decomposition of vNa $B$ into factors and if $P=\int\_{Y}^{\oplus} p\_ydy$ is a projection in $B$ and $p\_y$ is equivalent to a projection $q\_y$ in $B\_y$ for all or a.e. $y\in Y$. Is there a good choice of $q\_y$ up to equivalence such that $Q=\int\_{Y}^{\oplu... | https://mathoverflow.net/users/9401 | Disintegration of von Neumann algebra | Suppose B is a trivial bundle whose fibers are type I2 factors and p is a constant section of B corresponding to some projection with 1-dimensional image. Projections with 1-dimensional image in a type I2 factor can be identified with angles, i.e., elements of iR/πiZ. If q is given by some non-measurable section of B, ... | 3 | https://mathoverflow.net/users/402 | 78382 | 47,244 |
https://mathoverflow.net/questions/78385 | 5 | Is there a general way to decompose a complete graph $K\_n$ into an union of maximal matching subgraphs such that no two subgraphs share an edge?
For example, consider $K\_4$ with vertices $V=${1,2,3,4}. It can be decomposed into the union:
$K\_4=$ {{1,2},{3,4}} $\cup$ {{1,3},{2,4}} $\cup$ {{1,4},{2,3}}
In this e... | https://mathoverflow.net/users/18598 | Decomposition of a complete graph into maximal matching subgraphs | Let the $2n$ vertices be the $2n-1$ vertices of a regular polygon and its center. Join the center to one of the other vertices by a line segment $S$, then join the remaining $2n-2$ vertices in pairs by line segments perpendicular to $S$. That's one maximal matching subgraph. Now rotate the vertices around the center th... | 10 | https://mathoverflow.net/users/3684 | 78386 | 47,245 |
https://mathoverflow.net/questions/78239 | 4 | Some background to my question: if $G$ is a (simple) graph of $N$ vertices, labelled by integers $0,1,...,N-1$, the closeness centrality of a vertex $i$, denoted by $C(i)$, is defined to be the inverse of the mean distance of $i$ to all other vertices of $G$. (Here the distance $d\_{ij}=d\_{ji}$ between two vertices $i... | https://mathoverflow.net/users/15825 | Finding a vertex of least distance to all other vertices in a graph | A graph as you describe is normally called a tree, and yes, it is easy to compute the sum of distances to all other vertices in a tree.
First, choose one of the leaves (arbitrarily) to be the root of the tree, and then compute for each node the sum of the distances to its descendants, in a bottom-up traversal order, ... | 2 | https://mathoverflow.net/users/440 | 78391 | 47,247 |
https://mathoverflow.net/questions/78345 | 8 | Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F\_p[G]$?
Does there exists any book/paper where it is calculated?
By radical here I mean maximal ideal I of $F\_p[G]$ such that $I^n=0$
| https://mathoverflow.net/users/4246 | Radical of $F_p[SL(2,p)]$ | The answer depends a lot on what kind of description of the radical you ask for. This family of groups of Lie type has been well-studied from the viewpoint of modular representation theory in the defining characteristic (with reference also to the ambient algebraic groups). Even the somewhat degenerate case $p=2$ fits ... | 8 | https://mathoverflow.net/users/4231 | 78393 | 47,248 |
https://mathoverflow.net/questions/78402 | 11 | Does anybody know if the journal L'Intermédiaire des mathématiciens is on the web anywhere? Failing that, does anybody know of a library that has all or even some of the issues?
Thanks for any help.
Cheers, Scott
| https://mathoverflow.net/users/4111 | L'Intermédiaire des mathématiciens | Five minutes of googling resulted in the following page at the University of Michigan, which contains the full text of most volumes of this journal :
<http://mirlyn.lib.umich.edu/Record/009995560>
| 10 | https://mathoverflow.net/users/18602 | 78412 | 47,254 |
https://mathoverflow.net/questions/78404 | 30 | Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by elementary matrices, but I very much doubt that it is surjective in general (though I don't know any examples).
My ques... | https://mathoverflow.net/users/18602 | When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective? | A sort of universal example: Let $R$ be the polynomial ring $\mathbb Z[x\_{11},x\_{12},x\_{21},x\_{22}]$ and let $q$ be the ideal generated by $x\_{11}x\_{22}-x\_{12}x\_{21}-1$. The obvious element of $SL\_2(R/q)$ does not come from $SL\_2(R)$. You can see this by comparing with the example of the ring $\mathbb R[u,v]$... | 43 | https://mathoverflow.net/users/6666 | 78419 | 47,256 |
https://mathoverflow.net/questions/78415 | 6 | Given a set $E \subset \mathbb{R}^d$, define the distance set of $E$
$$
\Delta(E) = \lbrace|x-y| : x,y \in E \rbrace,
$$
where $|\cdot |$ is the usual Euclidean distance.
$\bullet$ The Erdos-distance conjecture roughly asserts that if a set $E \subset \mathbb{R}^d$ has finite cardinality $|E| = n$, then the distanc... | https://mathoverflow.net/users/15212 | Relationship between Erdos and Falconer distance problems | Generally speaking, continuous incidence geometry problems are considered strictly harder than their discrete analogues. For instance, if (for simplicity) one replaces Hausdorff dimension with Minkowski dimension (and glosses over the distinction between upper and lower Minkowski dimension), then the Falconer problem i... | 11 | https://mathoverflow.net/users/766 | 78425 | 47,258 |
https://mathoverflow.net/questions/78394 | 9 | A blog claims that the following Ramsey-type (or van der Waerden type) problem is open:
If the natural numbers are colored with finitely many colors, must there exist x and y (not both 2) such that x+y and xy are the same color?
Is it correct that this is an open problem, and can anybody help me track down a refere... | https://mathoverflow.net/users/12965 | Is this Ramsey-type problem an open problem? | The problem (and several extensions) was mentioned by Hindman.
The case of two colours was solved by Graham:
The interval $[1,252]$ contains $x$ and $y$ such that
$x,y, x+y$ and $xy$ are all monochromatic, and 252 is minimal.
References:
1) J Fox, Yeu-Whai Kathy Lin, and M Thibaul
The Clique Number of the ... | 15 | https://mathoverflow.net/users/7673 | 78427 | 47,260 |
https://mathoverflow.net/questions/78423 | 18 | How far can one go in proving facts about projective space using just its universal property?
Can one prove Serre's theorem on generation by global sections, calculate cohomology, classify all invertible line bundles on projective space?
I don't like many proofs of some basic technical facts very aesthetic because ... | https://mathoverflow.net/users/7041 | Clean Proofs of Properties of Projective Space | I think the answer is "probably not." The reason is that projective space has *two* universal properties which are used to prove different kinds of things about it. One of these is the slick universal property you like, and the other is the clunky one which results in unpleasantries.
Though each universal property im... | 22 | https://mathoverflow.net/users/1 | 78428 | 47,261 |
https://mathoverflow.net/questions/78422 | 11 | In his [first Eilenberg Lecture at Columbia](http://www.math.columbia.edu/~staff/EilenbergVideos/Gross/index.html#lecture_1), Benedict Gross says that only recently have we been able to give examples of finite galoisian extensions $K$ of ${\bf Q}$ which are ramified only at $2$ (respectively $3$) and for which the grou... | https://mathoverflow.net/users/2821 | Insolvable number fields ramified only at one (small) prime | Minhyong's comments indicate the issue here. If I want to come up with an extension unramified outside $p$ then why not look at the 2-dimensional mod $p$ representation attached to the $\Delta$ function? This works for all but a very small set of $p$, where either the mod $p$ representation is degenerate, or $p$ is so ... | 12 | https://mathoverflow.net/users/1384 | 78429 | 47,262 |
https://mathoverflow.net/questions/75961 | 13 | Let $U$ be a open affine subscheme of a smooth, proper scheme $X$ over $\mathbf{Z}\_p$. Over $\mathbf{Q}\_p$ we know that $\mathrm{H}^i(U \times \mathbf{Q}\_p/\mathbf{Q}\_p)$ is finite-dimensional (where $\mathrm{H}^i(\cdot/R) = \mathbf{H}^i(\Omega\_{\cdot/R}^\bullet)$ is the $i$-th de Rham cohomology of a scheme over ... | https://mathoverflow.net/users/10927 | Is de Rham cohomology of affine schemes over discrete valuation rings finitely generated (modulo torsion)? | After more studying, I found a solution. An $\overline{\cdot}$ will denote the reduction modulo $p$ of the object in question. Let us assume that $\overline{X}$ is a geometrically integral curve of genus $g$ and that $Z = X \setminus U$ is a (relative, reduced) normal crossing divisor. Then the following holds:
>
>... | 5 | https://mathoverflow.net/users/10927 | 78434 | 47,265 |
https://mathoverflow.net/questions/77814 | 2 | Take some stable theory $T$ with elimination of imaginaries, all sets appearing are small subsets of the monster model of $T$, elementary maps are restrictions of automorphism of the monster model of $T$.
Is the following statement correct?
1. For all algebraic closed sets $A,B$, with additionaly that these sets a... | https://mathoverflow.net/users/18449 | Extending elementary maps in stable theories. | Martin Ziegler provided this solution.
We got that $C$ is independent of $B$ over $A$. Then $C$ is independent of $acl(A\cup B)$ over $A$. So by stationarity (since acl(A)=A) we can extend $f$ to elementary $g:C\cup acl(A\cup B)\rightarrow C\cup acl(A\cup B)$ with $g\vert C=id$.
Since $C\cup acl(A\cup B)\subset ac... | 0 | https://mathoverflow.net/users/18449 | 78440 | 47,266 |
https://mathoverflow.net/questions/78437 | 1 | Let $G$ be a discrete group and K a subgroup of G . denote by $(\hat{H\_K})^i$ the bounded cohomology groups of $K$ , and by $(\hat{H\_G})^i$ the bounded cohomology groups of G.
then $(\hat{H\_K})^i$ is embedded in $(\hat{H\_G})^i$ ??
| https://mathoverflow.net/users/nan | bounded cohomology of subgroups of groups | In general, no. There is not even a natural map (in general) in the direction you want.
There is a natural map in the opposite direction, namely restriction, and this is sometimes an embedding, but not always. See chapter 8.6 in "Continuous bounded cohomology of locally compact groups", Lecture Notes in Mathematics 1... | 3 | https://mathoverflow.net/users/13877 | 78442 | 47,268 |
https://mathoverflow.net/questions/78435 | 2 | I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure.
| https://mathoverflow.net/users/4096 | Non-uniruled variety with level one Hodge structure. | I edited the answer to expand it and add more context:
The question was whether there exist nonuniruled smooth projective varieties with Hodge numbers $h^{pq}=0$ for all $|p-q|>1$. Of course, any curve of positive genus has this property. In dimension $2$, an Enriques surface, or any surface with $p\_g=0$ and nonnega... | 4 | https://mathoverflow.net/users/4144 | 78444 | 47,269 |
https://mathoverflow.net/questions/78443 | 5 | Let $F$ be a number field and $A$ an abelian variety over $F$. It is known that if $A$ has complex multiplication, then it has potentially good reduction everywhere, namely there exists a finite extension $L$ of $F$ such that $A\_L$ has good reduction over every prime of $L$.
And what about the inverse: if $A$ is kno... | https://mathoverflow.net/users/9246 | CM abelian varieties and potential good reduction | No, absolutely not
In fact, the hypotheses you discuss are rather weak. Take $F$ a totally real number field. If $A/F$ is the abelian variety attached to an eigenform $f$ of weight $2$ and level $N$, then the representation $\rho$ attached to the $p$-adic Tate module of $A$ is crystalline at $p$ for $p\nmid N$ . At $... | 4 | https://mathoverflow.net/users/2284 | 78447 | 47,270 |
https://mathoverflow.net/questions/78414 | 0 | The square of X which is $\aleph\_1$-calibre is still $\aleph\_1$-calibre?
| https://mathoverflow.net/users/18465 | $\aleph_1$-calibre | The answer is yes.
A topological space has *calibre* $\aleph\_1$ if for every uncountable sequence $\langle U\_\alpha\mid\alpha\lt\aleph\_1\rangle$ of nonempty open sets $U\_\alpha\subset X$, there is an uncountable subfamily $\Lambda\subset\aleph\_1$ with $\bigcap\_{\alpha\in\Lambda}U\_\alpha\neq\emptyset$.
**The... | 8 | https://mathoverflow.net/users/1946 | 78451 | 47,273 |
https://mathoverflow.net/questions/78397 | 7 | When one says that a stochastic process is Markovian, is this a property solely of the law of the process, or does the realization of the process also come in to play? I am asking even for the simplest examples, such as a process indexed by $\mathbb{N}$. Most abstract definitions are about being Markov with respect to ... | https://mathoverflow.net/users/9610 | Markov Property: determined by just the law or also the realization? | I've posted a solution [on my webpage](http://www.stat.ualberta.ca/people/schmu/preprints/markov_law.pdf).
| 4 | https://mathoverflow.net/users/nan | 78459 | 47,278 |
https://mathoverflow.net/questions/78450 | 23 | Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological spaces.
But the usual definition of singular homology is on the category of topological spaces, and you can show that it is... | https://mathoverflow.net/users/10217 | Homology theory constructed in a homotopy-invariant way | Throughout the following, I'll say "homotopy category" when I really mean the weak homotopy category.
For a space $X$, the homology of $X$ is canonically isomorphic to the reduced homology of $X\_+$, which is $X$ with a disjoint basepoint. Therefore, it suffices to give a definition of the reduced homology of a based... | 21 | https://mathoverflow.net/users/360 | 78466 | 47,283 |
https://mathoverflow.net/questions/78468 | 0 | The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
$$F(s)=\sum\_{1}^{\infty}\frac{f(n)}{n^{s}}.$$
The existence of an abscissa of absolute convergence $\sigma\_a$ naturally implie... | https://mathoverflow.net/users/10980 | Lower bounds for partial sums of multiplicative functions | If $f(n)$ is a multiplicative function then so is $f(n)n^{-w}$ for any fixed complex number $w$. In particular, find a multiplicative $f(n)$ for which $\sigma\_c$ is strictly smaller than $\sigma\_a$, and take $w=\sigma\_c-\epsilon$; the modified function $\tilde f(n) = f(n)n^{-w}$ will have its $\tilde\sigma\_c = \eps... | 5 | https://mathoverflow.net/users/5091 | 78477 | 47,287 |
https://mathoverflow.net/questions/78218 | 6 | Let $k$ be the field $F\_2((X,Y))$, where $F\_2$ is the field with two elements and
$X$ and $Y$ are two indeterminates. Can we describe the Brauer group of $k$, or at least its $2$-torsion?
(My motivation is as one could expect: I have an irreducible representation of a group of dimension 2 over the separable closur... | https://mathoverflow.net/users/9317 | Brauer group of a field of power series in two variables. | You can get a fairly good picture of the elements of order $2$ of the Brauer
group in the following way. There is no reason to fixate on characteristic $2$
so I assume that we are dealing with $K:=\mathbb F\_p((X,Y))$ and in fact the only
reason to stick to the prime field is notational convenience as the Frobenius
map... | 5 | https://mathoverflow.net/users/4008 | 78478 | 47,288 |
https://mathoverflow.net/questions/78471 | 47 | Let $D$ be a co-complete category and $C$ be a small category. For a functor $F:C^{op}\times C \to D$ one defines the *co-end*
$$
\int^{c\in C} F(c,c)
$$
as the co-equalizer of
$$
\coprod\_{c\to c'}F(c,c'){\longrightarrow\atop\longrightarrow}\coprod\_{c\in C}F(c,c).
$$
It is the indexed co-limit $\mbox{colim}\_W F$ whe... | https://mathoverflow.net/users/18595 | Intuition for coends | Martin's comment is right on the money; in particular, the best way to get a feeling for coends is through the many examples where they appear, such as (generalized) tensor products. But from an abstract point of view, coends can be considered as universal "[extranatural transformations](http://ncatlab.org/nlab/show/ex... | 45 | https://mathoverflow.net/users/2926 | 78484 | 47,289 |
https://mathoverflow.net/questions/78410 | 23 | Is there an example of a finitely presented infinite group in which every element has finite order? Or, is it known that every finitely presented infinite group has an element of infinite order?
I asked this question of math.stackexchange, thinking it might be trivial (for finitely generated groups there are numerous... | https://mathoverflow.net/users/6429 | Finitely presented infinite group with no element of infinite order? | This is just an expanded version of Igor's comment. Indeed this is an open problem. The common opinion (I believe) is that such groups do exist, but the best result in this direction so far is the Olshanskii-Sapir group, which is finitely presented and (infinite torsion)-by-cyclic.
There is a general idea, commonly ... | 38 | https://mathoverflow.net/users/10251 | 78489 | 47,291 |
https://mathoverflow.net/questions/78492 | 6 | Recall the fact that the representations of a quantum group form a *braided tensor category*, and this corresponds to the fact that $U\_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding operations can be constructed via the KZ equations for representations of $\mathfrak g$. The *first* thing that the KZ ... | https://mathoverflow.net/users/35353 | Kontsevich Integral without associators? | As you point out, the relation between associators and the quasi-triangular structure of $U\_q(\mathfrak g)$ (and the related tangles invariants) exists "only" at the Lie algebraic level, not (not yet) at the universal one. Roughly speaking, this is because the twisting which absorb the associator is not $\mathfrak g$-... | 10 | https://mathoverflow.net/users/13552 | 78499 | 47,295 |
https://mathoverflow.net/questions/78487 | 3 | Consider the complete graph on $n$ vertices. Each step, one chooses one of the $\binom{n}{2}$ edges iid uniformly at random. Say a sequence of choice is successful if there is some permutation of the vertices $[n]$, $i\_1,i\_2, \ldots, i\_n$, such taht the sequence contains a subsequence of the following form: $(i\_1,i... | https://mathoverflow.net/users/4923 | Bounding the success time of a coupon collector like problem | I think $n^4$ steps are sufficient: wait for (1,2) [ $n^2$ steps ]; then wait for (2,3); then (3,4); ... You're summing $n^2$ geometric random variables with parameter $1/\binom{n}{2}\approx 1/n^2$, so the expected time for success this way is $\Theta(n^4)$. Might you do better if you choose a different sequence? I don... | 2 | https://mathoverflow.net/users/11054 | 78502 | 47,297 |
https://mathoverflow.net/questions/78501 | 38 |
>
> Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point?
>
>
>
A *fixed point* for $G$ is a point $p \in B^n$ where for all $g \in G$ we have $g\cdot p = p$.
Notice that the answer is "yes" if $G$ is cyclic, by the Brouwer fixed poi... | https://mathoverflow.net/users/1650 | Do finite groups acting on a ball have a fixed point? | The answer is **no**.
A fixed point free action of the finite group $A\_5$ on a $n$-cell was constructed by Floyd and Richardson in their paper [An action of a finite group on an n-cell without stationary points](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-65/iss... | 50 | https://mathoverflow.net/users/7460 | 78504 | 47,298 |
https://mathoverflow.net/questions/78488 | 1 | Consider the heat equation $u\_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.
Is this true to say:
$$\|u(. , t)-v(. , t)\|\_p\leq \|u(. , 0)-v(. , 0)\|\_p$$ where $u$ and $v$ are two solutions of the heat equation in $W^{2,p}$.
| https://mathoverflow.net/users/18626 | Heat equation with Neumann BC | yes, with some regularity on the boundary.
Theorem 3.2.9 p. 90 of E.B. Davies book, Heat Kernels and Spectral Theory
gives Gaussian bounds for the heat kernel of an elliptic operator with Neumann boundary
conditions. These bounds imply that the heat flow preserves L^p.
| 1 | https://mathoverflow.net/users/18630 | 78514 | 47,304 |
https://mathoverflow.net/questions/78483 | 6 | Let $\mathfrak{g}$ be an Lie algebra of type G2. Are there some combinatorial ways to describe a basis of a $\mathfrak{g}$-module? For classical types, there is a method used tableaux. Thank you very much.
| https://mathoverflow.net/users/11877 | Reference request: representation of type G2 Lie algebras. | This paper constructs the representations of $G\_2$ explicitly in terms of the Schur functor construction and multilinear operations: <http://www.ams.org/journals/proc/1999-127-03/S0002-9939-99-04583-9/home.html>
It doesn't give you a preferred basis in terms of tableaux though. For that, you might want to look at th... | 9 | https://mathoverflow.net/users/321 | 78519 | 47,307 |
https://mathoverflow.net/questions/78526 | 4 | Suppose $f:\mathbb{C}\to\mathbb{C}$ is a map with your favorite smoothness condition (say, $C^1, C^{\infty}$ or holomorphic) and suppose that $f(\overline{\mathbb{Q}})\subset\overline{\mathbb{Q}}$. Is $f$ a polynomial? (I.e., if you chose $C^1$ or $C^{\infty}$ in the beginning, then is $f$ a polynomial in $z$ and $\ove... | https://mathoverflow.net/users/15630 | Maps preserving algebraic numbers | This is indeed very well-known. There are plenty of counterexamples. In fact, there's a remarkable construction due to van der Poorten of a "transcendental" entire function $f$ such that $f$ and all its derivatives map any algebraic number $\alpha$ into $\mathbb Q (\alpha)$. See
>
> A.J. van der Poorten, [Transcend... | 16 | https://mathoverflow.net/users/430 | 78527 | 47,312 |
https://mathoverflow.net/questions/78508 | 1 | Let $A$ be a complete discrete valuation ring with fraction field $K$ and perfect residue field $\kappa$.
Let $K\_{nr}$ be the maximal unramified extension of $K$ and let $A\_{nr}$ be its ring of integers.
Is there an easy way to see that $\text{Br}(A)$ is isomorphic to $H^2(\text{Gal}(K\_{nr}/K),A\_{nr}^\star)$?
... | https://mathoverflow.net/users/1107 | Brauer group of complete DVR | This is more trivial (in the sense that we have hidden all the non-trivial parts among general preliminaries...) than the identification of $\text{Br}(K)$ that Alex is talking about. We have that $\text{Spec } A\_{nr}\to \text{Spec } A$ is an algebraic universal covering map and $\text{Spec } A\_{nr}$ is acyclic so tha... | 8 | https://mathoverflow.net/users/4008 | 78531 | 47,316 |
https://mathoverflow.net/questions/78481 | 13 | I ran into this question on math.stackexchange.com "[How many 3 dimensional simple Lie algebras are there over the rationals?](https://math.stackexchange.com/questions/68521/three-dimensional-simple-lie-algebras-over-the-rationals)"
The question has been sitting idle for a long time. I thought it was interesting and wo... | https://mathoverflow.net/users/17263 | Three-dimensional simple Lie algebras over the rationals | Let me flesh out my comments. First, why there are infinitely many three-dimensional simple Lie algebras over $\mathbb Q$. One of the major steps in proving [class field theory](http://en.wikipedia.org/wiki/Class_field_theory) is to prove that we have an exact sequence of [Brauer groups](http://en.wikipedia.org/wiki/Br... | 14 | https://mathoverflow.net/users/6753 | 78535 | 47,319 |
https://mathoverflow.net/questions/78319 | 4 | Let $G$ be an infinite group such that for every $g\neq 1$ in $G$, there is some $b$ such that $C(b)$ has finite index in $G$ and $g$ is not in $C(b)$. Is something known about those groups? Are they FC groups, or related to FC groups?
| https://mathoverflow.net/users/18583 | Groups with trivial centralizer-connected component | Let $I$ be an infinite index set and for $i \in I$ let $G\_i$ be a finite group with trivial centre. Let $G:=\prod\_{i \in I} G\_i$ be the direct product of these groups.
Then $G$ satisfies the property you mentioned
$$ \forall g \in G\backslash \lbrace 1 \rbrace \exists b \in G: [G:C(b)]<\infty, g \not\in C(b) $$
but... | 2 | https://mathoverflow.net/users/3380 | 78536 | 47,320 |
https://mathoverflow.net/questions/78517 | 19 | I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C\*-algebras? If so, is there an analogue of the GNS construction?
| https://mathoverflow.net/users/10312 | P-adic C* algebras | There exists a complete theory of non-Archimedean commutative Banach algebras. In particular, there are conditions under which an algebra is isomorphic to the algebra of continuous functions. For the commutative case, they can be seen as the counterparts of the $C^\*$ condition. Note that there is no natural involution... | 27 | https://mathoverflow.net/users/12205 | 78538 | 47,321 |
https://mathoverflow.net/questions/78540 | 12 | Consider the standard Vandermonde $V(x\_1, \ldots, x\_n) = \prod\_{i < j} (x\_i - x\_j)$.
I am intersted in the calculation of the following expression for fixed $k$:
$$\sum\_i (x\_i)^k (d/dx\_i)^k V(x\_1 , \ldots , x\_n).$$
My guess is that it equals $c \cdot V(x\_1, \ldots, x\_n)$
where $c$ is an expression depending... | https://mathoverflow.net/users/10446 | A sum involving derivatives of Vandermonde | Perhaps a little Mathematica program will help us form a conjecture. For $k \geq n$ the answer is $0$, so we list your $c$ for $k < n$, as follows:
```
In[1]:= V[x_] := Product[x[[i]] - x[[j]], {i,1,Length[x]},{j,1,i-1}]
In[2]:= V[x/@Range[3]]
Out[2]= (-x[1]+x[2]) (-x[1]+x[3]) (-x[2]+x[3])
In[3]:= s[k_,x_] := Sum[x[... | 7 | https://mathoverflow.net/users/1176 | 78541 | 47,322 |
https://mathoverflow.net/questions/78542 | 9 | The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.
Its space of irreducible, continuous unitary representations has been described by Gelfand and Neumark in [this 1947 paper](http://www.ams.org/mathscinet-getitem?mr=20559).
I am not very familiar wi... | https://mathoverflow.net/users/10265 | Unitary representations of the ax+b group: an accessible presentation | I know of two clean approaches to classifying the unitary irreps of the $ax+b$ group. The first is to write the group as a semidirect product $\mathbb R \ltimes \mathbb R\_{>0}$. There is a theory (due chiefly to Mackey) that deals with the representations of semidirect products. In this case, the semidirect is fairly ... | 18 | https://mathoverflow.net/users/430 | 78545 | 47,325 |
https://mathoverflow.net/questions/78547 | 1 | I need to create about 100 (small) holes in a distributor plate (hole diam = 0.5 mm; plate diameter = 100 mm). The sm. holes should be distributed in such a way that the density (hole/area) is nearly constant, i.e. that the distance distribution function is close to monodispersed as possible (distance distribution = di... | https://mathoverflow.net/users/18639 | Constant hole density on the area of a circle | Circle packings in a circle may be what you want, although your criteria for evaluating configurations may be slightly different. For example, those studying circle packings find these [two configurations for N=13](http://www.buddenbooks.com/jb/pack/circle/n13.htm) equally good but you may prefer the configuration on t... | 2 | https://mathoverflow.net/users/2954 | 78548 | 47,326 |
https://mathoverflow.net/questions/78511 | 8 | Given a sequence of continuous functions $f\_n(x)$, all defined on a compact set $D$ and assuming $f\_n(x)$ is uniformly bounded. Let $f(x) = sup\_n f\_n(x)$.
It is clear that $f(x)$ is not necessarily continuous. For example, $f\_n(x) = 1-x^n, D=[0,1]$. But my questions is can $f(x)$ be discontinuous on a set with p... | https://mathoverflow.net/users/18629 | Can the supremum of continuous functions be discontinuous on a set of positive measure? | Given a closed set $E$, define the distance $d(x,E)$ from $x$ to $E$ in the usual way. Let $K\_n$ denote the set of $x$ so that $d(x,E)\ge 1/n$. Observe that the set $K\_n$ is closed and disjoint from $E$.
Urysohn's Lemma now says that there is a continuous function $f\_n:\mathbb{R}\rightarrow[0,1]$ which is 0 on $K... | 5 | https://mathoverflow.net/users/5751 | 78561 | 47,333 |
https://mathoverflow.net/questions/78361 | 8 | I guess one way of putting it, when does the series $\sum\_{x,y \in \mathbb{Z}} q^{x^2+xy+y^2}$ have nonzero coefficients?
The analogous answer for $\sum\_{x,y \in \mathbb{Z}} q^{x^2+y^2}$ is that $q^n$ appears when $\mathrm{ord}\_p(n)$ be even for all primes.
Is there a closed form for either of these with quadrat... | https://mathoverflow.net/users/1358 | which integers take the form $x^2 + xy + y^2$? | Let $r\_2(n)$ be the number of representations of $n$ as a sum of two squares, and let $l(n)$ be the number of ways to write $n$ as $x^2+xy+y^2$. Then as you mentioned we have that $$\sum\_{y,x\in \mathbb{Z}} q^{x^2+y^2}=\sum\_{n=0}^\infty r\_2(n) q^n \ \text{and} \ \sum\_{y,x\in \mathbb{Z}} q^{x^2+xy+y^2}=\sum\_{n=0}^... | 11 | https://mathoverflow.net/users/12176 | 78571 | 47,339 |
https://mathoverflow.net/questions/78574 | 9 | I have wondered for a while what gave rise to the notation $0^\sharp$. According to wikipedia this is due to Solovay in 1967, but (perhaps unsurprisingly) there's no discussion of why that notation was chosen. In contrast, Silver mentions in a paper that he chose the symbol $\Sigma$ to represent the same thing. Nonethe... | https://mathoverflow.net/users/18628 | What is the etymology of zero-sharp? | I've heard that the symbol was originally $O^\#$, with a capital letter O, not zero. This set was viewed as a higher-level analog of Kleene's O (a universal $\Pi^1\_1$ set).
| 10 | https://mathoverflow.net/users/6794 | 78582 | 47,343 |
https://mathoverflow.net/questions/78585 | 9 | ##Backgroud
The Nerve Theorem (see [nLab](http://ncatlab.org/nlab/show/nerve+theorem);) asserts that given a finite collection $\cal K$ of compact sets with the property that all non empty intersections of sets in the family are homotopically trivial, then the union $X$ of the sets in the family is homotopically equi... | https://mathoverflow.net/users/1532 | A Desirable Extension of the Nerve Theorem | The condition you suggest is insufficient. For example, let $F \to E \overset{\pi}\to B$ be a fibre bundle, with compact fibre say, (or more generally a local quasi-fibration), and $\mathcal{K}\_B$ be a cover of $B$ satisfying the conditions of the Nerve theorem (i.e. all intersections are empty or contractible). Then
... | 12 | https://mathoverflow.net/users/318 | 78588 | 47,345 |
https://mathoverflow.net/questions/78584 | 2 | Let me first recall the [Stone-von Neumann theorem](http://en.wikipedia.org/wiki/Stone%25E2%2580%2593von_Neumann_theorem) that if two one-parameter groups of unitary operators $U\_t$ and $V\_s$ over a Hilbert space satisfy $U\_tV\_s=e^{ist}V\_sU\_t$ for every $s,t\in{\mathbb R}$ (Weyl relations), then their generators ... | https://mathoverflow.net/users/8799 | omega-Commuting matrices vs Stone-von Neumann Theorem | I did not try to follow your argument, but here is another proof of your claim. In fact it proves a slightly stronger result, i.e. that if $\rho(1-U)$ and $\rho(1-V)$ are strictly smaller that $\sqrt 2$, $U,V$ cannot $\omega$-commute if $\omega \neq 1$.
First note that for a couple of unitaries $(U,V)$ that $\omega$-... | 4 | https://mathoverflow.net/users/10265 | 78593 | 47,348 |
https://mathoverflow.net/questions/78601 | 9 | First of all, i think MathOverflow is a very great community to discuss math, either basic or advanced, and i'm glad to participate here. It's my first post, so i'm sorry if i did anything wrong, and hope that people help me to make this post better. I also hope this post helps somebody with similar problem...
I've s... | https://mathoverflow.net/users/18651 | Binomial coefficient in Andrews' partition book | Consider the generating series
$$\sum\_{s=0}^{\infty}\left(\sum\_{i+j=s}\binom{A-n+j}{j}\binom{n-j}{i}\right)x^{s}.$$ This equals $$\sum\_{j=0}^{\infty}\binom{A-n+j}{j}x^{j}\sum\_{i=0}^{\infty}\binom{n-j}{i}x^{i}=(1+x)^{n}\sum\_{j=0}^{\infty}\binom{A-n+j}{j}\left(\frac{x}{1+x}\right)^{j},$$ where we use the binomial ... | 12 | https://mathoverflow.net/users/12176 | 78603 | 47,350 |
https://mathoverflow.net/questions/78406 | 0 | Is it true that a topology space X with a zeroset diagonal is first countable?
what if X is additionally CCC?
| https://mathoverflow.net/users/18465 | zeroset-diagonal | All it takes is a countable, not first-countable, Tychonoff space, say a countable dense subset, $D$, of the Cantor cube $2^{\mathfrak{c}}$. For every point $(d,e)$ off the diagonal there is a continuous function $f\_{(d,e)}$ on $D^2$ that is zero on the diagonal and has value $1$ at $(d,e)$. Now enumerate the compleme... | 2 | https://mathoverflow.net/users/5903 | 78604 | 47,351 |
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