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https://mathoverflow.net/questions/66567 | 10 | It is known that if a forcing notion is proper, then every P-filter will generate a P-filter in the generic extension (see, e.g., [Shelah, Proper and Improper Forcing, VI.5](https://projecteuclid.org/ebooks/perspectives-in-logic/Proper-and-Improper-Forcing/toc/pl/1235419814))
On the other hand, if we start collapsing... | https://mathoverflow.net/users/7281 | Destroying the P-filter-property | (A very partial answer.) Assuming CH, the answer seems to be "no" (for filters on countable sets).
CH implies that every P-filter on $\omega$ is generated by a tower (i.e., an almost decreasing sequence) of length $\omega\_1$. In any $\omega\_1$-preserving forcing extension any countable sequence of filter sets can b... | 8 | https://mathoverflow.net/users/14915 | 66885 | 41,220 |
https://mathoverflow.net/questions/66881 | 8 | Just a few days ago my seemingly eternal and recurrent fascination for Conway's *combinatorial game theory* (CGT) & surreal numbers had a recrudescence, so I grabbed this excellent [survey](http://arxiv.org/PS_cache/math/pdf/0410/0410026v2.pdf), and began reading.
Some old thoughts came to the surface from the archi... | https://mathoverflow.net/users/15293 | Sets as Combinatorial Games | I think the answer to the first question (in bold) is, "of course." Just as, quoting On Numbers and Games,
* If $L, R$ are any two sets of numbers, and no member of $L$ is $\geq$ than any member of $R$, then there is a number $\{L|R\}$. All numbers are constructed this way
we can say
* If $L, R$ are any two set... | 7 | https://mathoverflow.net/users/2926 | 66887 | 41,221 |
https://mathoverflow.net/questions/66520 | 7 | Let $s\_0$ and $s\_1$ be the holomorphic sections of the tautological bundle $O(1)$ over the complex projective line ${\mathbb{CP}}^1$ which correspond to the functions $1$ and $\frac{x\_1}{x\_0}$ in the open set $U\_0= \{x\_0\neq 0\}$. Let $U(z,w)=s\_0(z)\overline{s\_0(w)} + s\_1(z)\overline{s\_1(w)}$ be the bilinear ... | https://mathoverflow.net/users/nan | Geometric interpretation of the argument of the Fubini-Study bilinear form on projective space? | I'm not sure what you mean by complex-angular measure, but there is indeed a geometric interpretation.
Suppose $z$ and $w$ have homogeneous coordinates $[z\_0,z\_1]$ and $[w\_0,w\_1]$, respectively. Then
$$
U(z,w) = 1 + \frac{z\_1}{z\_0} \frac{\overline{w}\_1}{\overline{w}\_0},
$$
and hence
$$
\widetilde{U}(z,w) = \f... | 5 | https://mathoverflow.net/users/4720 | 66889 | 41,223 |
https://mathoverflow.net/questions/66860 | 4 | What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know:
1. Is $K(\zeta\_n)/K$ an abelian extension for every $n$?
2. What are the roots of unity in the ray class field of $K$ with conductor $\mathfrak{c}$?
... | https://mathoverflow.net/users/434 | What are the roots of unity in abelian extensions of imaginary quadratic fields? | Just as a minor warning: even if the conductor is $1$, there might be nontrivial roots of unity in the class field: take $K = {\mathbb q}(\sqrt{-5}\,)$ and ${\mathfrak c} = (1)$;
then the ray class field is the Hilbert class field $K(\sqrt{-1})$, which contains the 4th roots of unity. The roots of unity in the Hilbert... | 3 | https://mathoverflow.net/users/3503 | 66900 | 41,227 |
https://mathoverflow.net/questions/66683 | 9 | **Question:** I am talking about the proof given on pages 50-52 of Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten (editors), *Quantum Fields and Strings: A Course for Mathematicians*, Volume 1, AMS 1999 (on [google books](http://books... | https://mathoverflow.net/users/2530 | I don't get a part of Bernstein's / Deligne-Morgan's proof of Poincaré-Birkhoff-Witt | Here is an explanation Pavel Etingof has given to me in email. Thanks Pavel!
Every $\sigma\in S\_{n}$ satisfies
$\sum\limits\_{i}\left( n-i+1\right) y\_{\sigma1}\ast\cdots\ast\left[ x,y\_{\sigma i}\right] \ast\cdots\ast y\_{\sigma n}$
$=n\left[ x,y\_{\sigma1}\right] \ast y\_{\sigma2}\ast\cdots\ast y\_{\sigma n}$
... | 5 | https://mathoverflow.net/users/2530 | 66909 | 41,231 |
https://mathoverflow.net/questions/66917 | 31 | Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for exams and because I was lead to it through a very convoluted path of research (from arithmetic through algebraic geomet... | https://mathoverflow.net/users/1985 | Classification of PDE | PDE books often discuss classification, but they always restrict attention to the case of second order equations, especially for one function of several variables, with good reason. The point of a classification is to find categories of PDE whose analysis has many common features, but there really isn't any general cla... | 17 | https://mathoverflow.net/users/13268 | 66920 | 41,239 |
https://mathoverflow.net/questions/66925 | 2 | **Question:** Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-module along with a $\mathbb Z$-linear map ${}^{[p]}:\mathfrak g\to\mathfrak g$ (written postfix) that satisfies $\lef... | https://mathoverflow.net/users/2530 | Restricted universal enveloping algebra of Abelian p-Lie algebra | I don't think so. Consider the case which should be the most difficult to split canonically, the case when the $p$'th power map is zero. The automorphism group is then equal to the linear automorphism group of $\mathfrak g$ and I assume further that $k$ is an infinite field and $\mathfrak g$ a finite dimensional vector... | 4 | https://mathoverflow.net/users/4008 | 66934 | 41,245 |
https://mathoverflow.net/questions/66865 | 35 | Let $k$ be a field, let $G = PGL\_2(k)$ be the projective general linear group of $k$, and let
$X = k \cup \{ \infty \}$ be one-dimensional projective space over $k$. Then $G$ acts on $X$ (via fractional linear transformations). This action has the following properties:
1) The action of $G$ on $X$ is simply 3-transit... | https://mathoverflow.net/users/7721 | Action of PGL(2) on Projective Space | A KT-field $(F,+,\times,\sigma)$
consists of a neardomain $(F,+,\times)$ together with an involutionary
automorphism $\sigma$ satisfying
$$\sigma(1 + \sigma(x)) = 1 - \sigma(1 + x)$$
for all $x \in F \setminus \{0,1\}$. (My impression is that neardomains are quite weak entities, e.g. $F^{\times}$ is required to be a gr... | 28 | https://mathoverflow.net/users/nan | 66936 | 41,246 |
https://mathoverflow.net/questions/66937 | 0 | Hey guys, I recently stumbled across this interesting sequence:
1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145 ...
Any ideas? The sequence appears to somewhat resemble the binary sequence.
| https://mathoverflow.net/users/15595 | What is the pattern in this sequence of integers | I would try this: <http://oeis.org/A034729>
$$ a(n) = \sum\_{k, k|n } 2^{(k-1)}$$
In Mathematica syntax this is (see [W|A](http://www.wolframalpha.com/input/?i=Table%5BSum%5B2%5E%28k-1%29,%7Bk,Divisors%5Bn%5D%7D%5D,%7Bn,1,20%7D%5D))
```
Sum[2^(k - 1), {k, Divisors[n]}]
```
| 4 | https://mathoverflow.net/users/358 | 66938 | 41,247 |
https://mathoverflow.net/questions/66890 | 1 | In a paper I am reading at the moment (Hrushovski Martin, elimination of imaginaries in $Q\_p$), in some proof they use the following fact (at least this would be enough to get their proof going, but maybe we have more hypothesis) :
If we have a group $G$ acting on a set $X$ containing elements $(e\_i\mid i<\omega)$ ... | https://mathoverflow.net/users/15587 | A problem with Neumann's lemma. | Let $S\_\infty=\bigcup\_{n=1}^\infty S\_n$ be the infinite symmetric group consisting of permutations of $\mathbb{N}$ fixing all but finitely many elements. Then $\{e\_i=i\}$ satisfy your conditions but no element $i\in \mathbb{N}$ is fixed by $S\_\infty$.
(Sorry, I see that Andreas Blass already gave this counterexa... | 0 | https://mathoverflow.net/users/250 | 66942 | 41,249 |
https://mathoverflow.net/questions/66941 | 2 | If $f:\mathbb{N}\to\mathbb{N}$ is any strictly increasing function with $f(0)=1$, define the *base $f$* notation for natural numbers inductively as follows:
1. $0$ is represented as $()$ (the empty sequence).
2. If $n>0$, the representation of $n$ is defined as follows. Let $k$ be maximal such that $f(k)\leq n$. The ... | https://mathoverflow.net/users/15455 | Here is a generalization of n-ary base notation for numbers. Surely unoriginal. Anybody know where to find literature on it? | Essentially the system you describe is given at the beginning of section 2 of the paper:
"Systems of Numeration" by Aviezri S. Fraenkel
in The American Mathematical Monthly, Vol. 92, No. 2 (Feb., 1985), pp. 105-114.
See: <http://www.jstor.org/pss/2322638>
Other systems are described as well.
| 6 | https://mathoverflow.net/users/7222 | 66947 | 41,251 |
https://mathoverflow.net/questions/66952 | 2 | I believe the following statement is true:
Given a complex analytic map $f:\Delta\to V/G$, where $\Delta$ is a disc in $\mathbb{C}$, $V$ a finite dimensional complex vector space and $G$ a finite subgroup of $GL(V)$, then $f$ admits an analytic lift $\tilde f:\Delta'\to V$ up to a ramified cover. More precisely, ther... | https://mathoverflow.net/users/15600 | Lifting analytic map | Salut Yann!
You can refer to the general notion of *fiber product*, for example. This is a construction that works in the complete generality. It might happen that in the situation you consider the fiber product will not be irreducible, (this happen when the preimage of $f(\Delta)$ in $V$ is not irreducible), then yo... | 4 | https://mathoverflow.net/users/943 | 66953 | 41,255 |
https://mathoverflow.net/questions/66956 | 3 | Given a bunch of boolean variables $a\_i \in \{0, 1\}$,
I want to write a boolean formula to express $\sum\_{i=1}^n a\_i \geq k$.
i.e. I'm allowed to use $(, ), \wedge, \vee, \lnot$.
Now, if I allow the use of $\exist$, then I can do this as a formular of length O(n^c) (basically create a circuit that adds together... | https://mathoverflow.net/users/3609 | Expressing >= as a boolean formula. | Yes, there exist (uniformly constructible) polynomial-size Boolean formulas for threshold functions (which is how your functions are called). Equivalently, there are polynomial-size formulas for summing $n$ binary numbers of length $m$. Also equivalently, the complexity class (uniform) $\mathrm{TC}^0$ is contained in (... | 6 | https://mathoverflow.net/users/12705 | 66966 | 41,259 |
https://mathoverflow.net/questions/66943 | 11 | I'm recently working on something called *3d mirror symmetry* in QFT literature, which involves two hyperkähler manifolds.
There seems to be a corresponding(?) mathematical theory called *symplectic duality*, pursued by Braden, Licata, Proudfoot and Webster.
Where can I read about it? The only thing I could find so... | https://mathoverflow.net/users/5420 | What is the "symplectic duality" between holomorphic symplectic manifolds? Where can I read more about it? | Nowhere. The paper is still in preparation, and looks to be for a few more months at least. Probably the best document at the moment is this (extremely long) set of [talk slides](http://pages.uoregon.edu/bwebster/austin-dual.pdf) of mine.
I should note: even when there is a paper, there won't be a definition that wil... | 7 | https://mathoverflow.net/users/66 | 66967 | 41,260 |
https://mathoverflow.net/questions/66984 | 4 | I am looking for a reference to study logarithm of an invertible triangular matrix. What is a good algorithm? Are there any good reference which studies this topic both theoretically and from an algorithm view point? I am also looking for structure of the logarithm of an upper/lower triangular invertible matrix.
| https://mathoverflow.net/users/10035 | Logarithm of a matrix | You may want to take a look at [*"Functions of Matrices: Theory and Computation"*](http://books.google.com/books?id=S6gpNn1JmbgC&printsec=frontcover&dq=functions+of+matrices&hl=en&src=bmrr&ei=2ePrTfX3E8bFswa6_d3nCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false) by Higham. Chapter 11 is spec... | 12 | https://mathoverflow.net/users/5371 | 66985 | 41,271 |
https://mathoverflow.net/questions/66973 | 5 | I need to compute a large number of inverses of the following form:
$(A \Lambda\_k A^\top)^{-1}$
where $A \in \mathbb{R}^{m \times n}$, $n > m$ and $\Lambda\_k = \text{diag}(\lambda\_1, ..., \lambda\_n)$ with $\lambda\_i > 0$. Is there an efficient way to do this?
In the end, I want to sample from Gaussians with ... | https://mathoverflow.net/users/15603 | Inverting products of matrices | Now that you've provided some more information, I think I can make some useful suggestions.
First, a quick review of linear transformations of multivariate normal random vectors. If $z$ is an MVN vector with mean $\mu$ and covariance matrix $C$, and $M$ is a matrix of the appropriate size and $y=Mz$, then $y$ is MVN... | 3 | https://mathoverflow.net/users/9022 | 66986 | 41,272 |
https://mathoverflow.net/questions/2179 | 15 | I wonder if the following is known: Are there two compact curves C1 and C2 of genus>1 defined over complex numbers, such that their product contains infinite number of irreducible curves of negative self-intersection and arbitrary large genus? It we aks the same question replacing "negative" by "zero", the answer will ... | https://mathoverflow.net/users/943 | Curves with negative self intersection in the product of two curves | **Disclaimer.** The answer below is a variation of Bogomolov's argument and it would not come to be without Dmitri's answer. If you feel like upvoting this, please upvote his answer.
**Curves on products of isogeneous elliptic curves.**
As already suggested in the body of the question, if we start with a pair of ell... | 7 | https://mathoverflow.net/users/605 | 67008 | 41,285 |
https://mathoverflow.net/questions/67003 | 3 | For $k, n \in \mathbb{N}$, let $\mathcal{C}\_n \mathbb{R}^k$ denote the configuration space of $n$ distinct points in $\mathbb{R}^k$.
* (1) Is there a description of the tangent space $T\_C \mathcal{C}\_n \mathbb{R}^k$ in terms of the configuration $C$?
Equipping $\mathbb{R}^k$ with the usual metric, a regression ... | https://mathoverflow.net/users/6862 | Least-squares regression and differential geometry | By calculus, the line $l\_C$ is 'the' major axis of the ellipse of inertia of the finite point set $C$. (The reason for the quotes around 'the' in the previous sentence is that, if the ellipse of inertia is a circle, then $l\_C$ is not well-defined; any line through the center of mass will minimize $E\_C$.) I'm not sur... | 5 | https://mathoverflow.net/users/13972 | 67012 | 41,287 |
https://mathoverflow.net/questions/67019 | 4 | The answer is certainly "Yes", but this is the problem I met in *Algebraic Number Theory* by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are no totally ramified extensions except the trival ones".
Let $K$ be Henselian field, $L/K$ be a finite, totally ramifi... | https://mathoverflow.net/users/15139 | Does totally ramified extension really exist? | Henselian-ness means that any factorisation of $\overline{f}$ into *coprime* monic polynomials in $\kappa[X]$ lifts to $K[X]$. But the factorisation of $\overline{f}$ could be a power of an irreducible polynomial, and this is indeed what happens in practice: consider $K = \mathbb{Q}\_p$ and $L = K[x]$ where $x^2 = p$. ... | 10 | https://mathoverflow.net/users/2481 | 67022 | 41,293 |
https://mathoverflow.net/questions/66977 | 9 | I am reading Veronique Godin's famous article "Higher string topology operations" (<http://arxiv.org/abs/0711.4859>) that demonstrates that the string topology operations on $(H\_\bullet(L X), H\_\bullet(P X))$ for $X$ a compact oriented smooth manifold are part of a homological 2d TQFT with target space $X$.
The ide... | https://mathoverflow.net/users/381 | Pull-push in Godin's HCFT for string topology | Urs, although I answered some of these questions privately, I will post them here as well.
At the moment there is no technology to do the naive short zig-zag you suggest. Godin's techniques for umkehr maps of mapping spaces of maps into a manifold $M$ can do essentially two things: (1) create isolated points in the d... | 9 | https://mathoverflow.net/users/798 | 67023 | 41,294 |
https://mathoverflow.net/questions/67027 | 4 | I want to learn the book, but it seems that I should have some background on harmonic analysis, Lie Groups and measure theory. Can you give some references?
| https://mathoverflow.net/users/3525 | Preliminaries for Mumford's Abelian Varieties | To discuss generally first, the book was written up by C. P. Ramanujam, and he was more conscientious than usual in trying to tie down Mumford's lectures to existing references. Still, it is quite hard to sort out the exact prerequisites.
The analytic theory and theta-functions required in the first chapter stands r... | 5 | https://mathoverflow.net/users/6153 | 67031 | 41,297 |
https://mathoverflow.net/questions/66929 | 6 | Given a trivalent tree (graph without loops with valence of 3+ at each vertex) on N marked points, let's assign to each vertex the number (its valence - 3)! (note the ! at the end). Take the product of these numbers over all vertices in the graph. Sum the resulting products over all trivalent graphs on N marked points.... | https://mathoverflow.net/users/27140 | number of weighted trivalent trees | This is just building off the previous discussion -- I will just give some indication of why Dan's $f$ and $g$ are Legendre transforms of each other. This is essentially a fleshed out version of some of what Frédéric was saying that I wanted to work out in detail -- it would look too small and ugly as a comment.
We'... | 3 | https://mathoverflow.net/users/1102 | 67034 | 41,300 |
https://mathoverflow.net/questions/67042 | 1 | Let $Q\to S$ be a quadric fibration over a rational base $S$, over an algebraically closed field of non zero characteristic. Is it true the following?
$Q$ is rational if and only if $Q \to S$ has a rational section.
If not, may it be true under some assumptions (bounds on the dimensions, on the associated Clifford ... | https://mathoverflow.net/users/15617 | Rationality of quadric fibrations | In one direction the implication is evident --- if there is a section then $Q$ is rational.
In the other direction the implication is false. For example, consider any projective space $P(V)$, let $S = P(S^2V^\*)$, and $Q$ be the universal quadric, that is the canonical divisor of bidegree $(2,1)$ on $P(V)\times P(S^... | 2 | https://mathoverflow.net/users/4428 | 67062 | 41,312 |
https://mathoverflow.net/questions/67039 | 6 | Two questions concerning the decomposition of $L\_0^2(GL\_2({\mathbb{Q}}) \backslash GL\_2(A), \psi)$, where $\psi$ is a Hecke character on the adelic ring $A$:
1. It is known that when $\psi$ is trivial on $\mathbb{R}^\times\_{+}$, there is an explicit correspondence between classical modular forms and irreducible c... | https://mathoverflow.net/users/14037 | Decomposition of $L_0^2(GL_2({\mathbb{Q}}) \backslash GL_2(A), \psi)$ | Originally, I wrote: Assuming that I understand the intent of the question properly, it is intended that psi be a Hecke character on the center of GL(2, adeles). The question seems to grant that we understand the situation with non-trivial Hecke characters subject only to the requirement that they be trivial on the "ra... | 12 | https://mathoverflow.net/users/15629 | 67064 | 41,313 |
https://mathoverflow.net/questions/64572 | 3 | Hi, Everyone:
I would appreciate some references for the version of Reidemeister-Schreier that is used to find the stabilizer of a point under a group action. The only refs. I have found
are about Schreier-Sims method, but I have
not been able to find anything on it.
The version of R-S I know of allows us to find... | https://mathoverflow.net/users/15017 | Reidemeister-Schreier Method for Finding Stabilizer of an Element in a Group Action | This is an answer to the OP's second question. Let $H$ be a group acting, say on the right, on a set $S$. Suppose that $H$ is generated by $X$, and let $G$ be the graph with vertex
set $S$ and edges of the form $(s,sx)$, where $s \in S$ and $x \in X$. Then $G$ is connected if and only if $H$ acts transitively on $H$: a... | 4 | https://mathoverflow.net/users/7021 | 67066 | 41,314 |
https://mathoverflow.net/questions/67063 | 7 | Let $G$ be a reductive connected algebraic group and let $B$ a Borel subgroup. One of central themes of the representation theory of $G$ is the study of the induction functor $H^0$ from $B$ representations to $G$ representations. Many of the features of $H^0$ in the characteristic zero case also hold in the modular cas... | https://mathoverflow.net/users/40886 | Kempf Vanishing theorem and Representation of Lie algebras. | As usual the history is a little complicated to track, but it should be understood first that the term *Weyl module* and notation $V(\lambda)$ were first used to describe the module obtained by a natural reduction modulo $p$ process from the usual finite dimensional simple module (of dominant highest weight $\lambda$).... | 12 | https://mathoverflow.net/users/4231 | 67072 | 41,317 |
https://mathoverflow.net/questions/66803 | 7 | Let $X$ be a path-connected nilpotent space (meaning $\pi\_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi\_1(X)\to \mathrm{Gl}(V)$ be a representation, where $V$ is a rational vector space. Then $\rho$ defines a local coefficient system on $X$, and we have the cohom... | https://mathoverflow.net/users/8103 | Minimal models with local coefficients | I think one could argue as follows: one can start by constructing a cochain complex $A^\*(X,\rho)$ that computes the twisted cohomology so that it will be a module over the Sullivan cochains $A^\*(X)$ of $X$ (with constant coefficients). Then one can plug it in Hinich's machinery: see <http://arxiv.org/PS_cache/q-alg/p... | 3 | https://mathoverflow.net/users/2349 | 67076 | 41,321 |
https://mathoverflow.net/questions/67045 | 11 | Are there are any good lower bounds for the number of spanning trees for a *connected* graph $G$ in terms of (for example) number of edges $E$ or number of vertices $V$ ?
Are improved bounds available if one knows the number of *bridges* of the graph? (A bridge, aka a cut-edge, is an edge whose removal disconnects t... | https://mathoverflow.net/users/9896 | Lower bound on # spanning trees in a connected graph | A quick google turns up:
[Undirected simple connected graphs with minimum number of spanning trees](http://www.sciencedirect.com/science/article/pii/S0012365X08005050) by Zbigniew R. Bogdanowicz. According to this paper, the optimal graph is built as follows: Start with the complete graph $K\_{n-k}$. Take $k-1$ addit... | 10 | https://mathoverflow.net/users/297 | 67087 | 41,327 |
https://mathoverflow.net/questions/67046 | 10 | Perelman's stability theorem shows in particular that a finite dimensional compact Alexandrov space $(X,d)$ such that $X$ is not a topological manifold cannot be approximated in the Gromov-Hausdorf topology by Riemannian manifolds of the same dimension whose sectional curvature is bounded from below.
My questions are... | https://mathoverflow.net/users/8887 | Metrically singular Alexandrov space. | **Suspicious example:**
Take a "funny" manifold with sectional curvature $\ge 1$ say $X$;
funny means Cayley flag or Aloff--Walach/Eschenburg/Bazaikin space, (not $S^n$ or $\mathbb{C}\mathrm{P}^n$ or $\mathbb{H}\mathrm{P}^n$).
Consider spherical suspension $\Sigma$ over $X$.
The space $\Sigma$ has curvature $\ge 1$.
... | 9 | https://mathoverflow.net/users/1441 | 67090 | 41,330 |
https://mathoverflow.net/questions/67041 | 3 | [This is a complete rewrite which makes some of the comments redundant or irrelevant.]
Take a set of $50$ elements. How many subsets of size $5$ are needed so that every subset of size $5$ will intersect at least one of these in at least $2$ points?
This collection of subsets is known as a lottery wheel or a [lott... | https://mathoverflow.net/users/15616 | How many elements with a hamming distance of 3 or less? | You are not using the positions at all. You have 50 points. $S$ is the set of all $\binom{50}{5}=2118760$ selections of 5 points. You want a subset $B \subset S$ such that any $s \in S$ intersects at least one $b \in B$ in at least 2 points. That is an interesting problem and does not immediately strike me as familiar.... | 7 | https://mathoverflow.net/users/8008 | 67104 | 41,336 |
https://mathoverflow.net/questions/67093 | 13 | [Carleson theorem](http://www.ams.org/mathscinet-getitem?mr=0199631) (later extended by Hunt) states that given an $L^2$ function $f:{\mathbb R}/{\mathbb Z}\to{\mathbb C}$, the set of points $x$ where the Fourier series $$\lim\_{n\to\infty}\sum\_{k=-n}^n\hat f(k)e^{2\pi ik x}$$ does not converge to $f(x)$ has measure 0... | https://mathoverflow.net/users/6085 | Sets of divergence of Fourier series | I believe that the problem of characterizing the sets of divergence for classical Fourier series is more or less open for all interesting classes ($C$, $L^\infty$, $L^p$ with $p>1$).
The strongest result that I'm aware of is due to Buzdalin who showed that any null-set $E\in F\_\sigma\cap G\_\delta$ is a set of diver... | 5 | https://mathoverflow.net/users/5371 | 67116 | 41,343 |
https://mathoverflow.net/questions/67102 | 1 | In the Princeton Companion of Mathematics, the Analytic Number Theory section, the author mentions what he calls Gauss-Kramer model, which is simply modeling the integers on a countable sequence of random variables $\{ X\_{n} \}$, where the variable $X\_{n}$ has probability of $1/log(n)$ of being 1 and otherwise 0. Eac... | https://mathoverflow.net/users/3873 | Is anyone aware of a good exposition of the Gauss-Kramer model of Integers? | Chapter 3 of The Prime Numbers and Their Distribution by Gérald Tenenbaum and Michel Mendès France has a nice exposition of the model, including modifications indicated by Maier's discovery.
| 1 | https://mathoverflow.net/users/13218 | 67122 | 41,346 |
https://mathoverflow.net/questions/66642 | 4 | Let $S$ be a $K3$ surface and $H$ be an ample line bundle on it. Given a flat family of coherent sheaves on $S$ whose generic point is a $\mu\_H$-stable vector bundle, what can I say about the non-generic points? Are they always $\mu\_H$-semistable and torsion free? I think that this is equivalent to requiring the exis... | https://mathoverflow.net/users/33841 | Limit of stable vector bundles. | I think the answer to your question is **no**, because of the following example. For the details, see [Huybrechts-Lehn, The geometry of moduli space of sheaves, Section 5.3].
Let $\pi \colon X \to \mathbb{P}^1$ be an elliptic $K3$ surface with irreducible fibres and assume that there is a section $\sigma \subset X$. ... | 2 | https://mathoverflow.net/users/7460 | 67125 | 41,348 |
https://mathoverflow.net/questions/67077 | 8 | Given a topological space $X$, form the category $\mathrm{Cov}(X)$ consisting of open subsets $U \subset X$ as objects and inclusions as morphisms. To what extend can we recover (properties of) $X$ from $\mathrm{Cov}(X)$?
Example: $\mathrm{Cov}(X)$ determines the Cech cohomology groups of $X$, and so for nice $X$ its... | https://mathoverflow.net/users/nan | to what extent does the category Cov(X) determine a topological space X? | In order to have enough freedom, I would prefer to consider a basis $\mathcal{U}$ of open subsets of $X$. In this case, considering $\mathcal{U}$ as a partially ordered set (for the inclusion), there is a canonical inclusion functor into topological spaces
$$\mathcal{U}\to \mathit{Top} \ , \ U\mapsto U$$
whose coli... | 11 | https://mathoverflow.net/users/1017 | 67127 | 41,349 |
https://mathoverflow.net/questions/67101 | 3 | Hello,
If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ times along a disc bounding $U$. How can I visualize $p/q$ dehn filling?. I mean, in that case after you do the filling you ... | https://mathoverflow.net/users/7894 | A question about Dehn filling in the unknot. | The complement of $U$ is a copy of $S^1\times D^2$. Let $\widetilde V$ be the $p$-fold cover of $V$ in the $p$-fold cover of $S^1\times D^2$. Cut $\widetilde V$ along $pt \times D^2$, twist $q$ times, then reglue to get a link $\widetilde V\_t$ in $S^1\times D^2$. Embed the modified $S^1\times D^2$ back into $S^3$ in t... | 3 | https://mathoverflow.net/users/284 | 67135 | 41,353 |
https://mathoverflow.net/questions/67133 | 3 | On the TV channel SBS, in Australia, there is a TV show in which contestants have six numbers and the operations of addition, subtraction, multiplication and division with which to produce a three digit number.
My question is whether, for any 6 numbers, this is always possible, and if so, does it hold for any choice ... | https://mathoverflow.net/users/15638 | "Letters and Numbers" Numbers game | I assume the six numbers are all distinct, and each must be used exactly once. A brute-force program found, e.g. that with the six numbers 4, 6, 8, 16, 32, 64 the possible results did not include 571, 581, 587, 619, 623, 631, 649, 657, 661, 671, 673, 679, 681, 695, 709, 713, 721, 731, 743, 793, 811, 817, 821, 823, 827,... | 4 | https://mathoverflow.net/users/13650 | 67141 | 41,355 |
https://mathoverflow.net/questions/67137 | 0 | I want to verify the following claim that I found in some paper. Suppose f is a smooth real-valued function on the real line satisfying $f'(x)x-f(x)\ge x^2$ for all x. Then there is a constant C, s.t. $f(x)\ge \frac{x^2}{2}-C$ for all $x.$
The connection with the title is that the authors claim that any function sati... | https://mathoverflow.net/users/3509 | on a property of functions of quadratic growth | Note that
$$\left( \frac{f(x)}{x} \right)' = \frac{xf'(x) - f(x)}{x^2} \geq 1,$$
hence the result.
| 4 | https://mathoverflow.net/users/14037 | 67142 | 41,356 |
https://mathoverflow.net/questions/67123 | 2 | I would like to find all integer triples (x,y,z) such that: $\prod\_{\theta}(x + y \theta + z \theta^2)=1$, where $\theta$ runs through the solutions to the cubic $x^3 + x^2 - 2x - 1=0$.
In his book "Diophantine equations" (p. 111-12) Mordell gives the equation
$w^n = \prod\_{\theta}(x + y \theta + z \theta^2)$
... | https://mathoverflow.net/users/15635 | representation of integers as the product of linear forms in three variables | This isn't really a research-level question, and hence belongs more on math.SE than here, but here goes anyway...
Let $K$ be the number field $\mathbb{Q}(\theta)$ where $\theta$ is a root of your cubic $f$. Then the ring of integers $\mathcal{O}\_K$ of $K$ is $\mathbb{Z}(\theta)$, and we are reduced to looking for el... | 6 | https://mathoverflow.net/users/2481 | 67148 | 41,359 |
https://mathoverflow.net/questions/67154 | 4 | In every $n$-category (weak or strict) can be defined the concept of equivalence via a recursive definition:
\* an equivalence in a set ($0$-category) is just an identity;
\* for each $n \in \mathbb N$ an equivalence between two object (or $0$-cell), let say $a$ and $b$, in a $n+1$-category is just a $1$-cell $f \colon... | https://mathoverflow.net/users/14969 | Equivalence in $\infty$-categories | For some precise definitions and results, see this [paper](http://www.cheng.staff.shef.ac.uk/duals/duals1103.pdf) by Eugenia Cheng.
| 7 | https://mathoverflow.net/users/2926 | 67159 | 41,367 |
https://mathoverflow.net/questions/67152 | 8 | I have a decision problem that I have formulated as a feasibility SDP. The answer to the decision problem depends on whether the SDP is feasible or not. It is known that a SDP can be solved to arbitrary precision in time that is polynomial in the input size as well as the precision value. Now can I claim that I have a ... | https://mathoverflow.net/users/39663 | SDP Feasibility | No, you can't. If you could, then you would, among other things, have a solution to what is known as the [square root problem](http://garden.irmacs.sfu.ca/?q=op/complexity_of_square_root_sum): given $a\_1, \ldots, a\_n,k$ determine whether $ \sum\_{i=1}^n \sqrt{a\_i} \leq k.$ This looks like a solvable problem - just c... | 10 | https://mathoverflow.net/users/9316 | 67160 | 41,368 |
https://mathoverflow.net/questions/67118 | 7 | Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of ZF-Infinity in PA (see for refs this MO [question](https://mathoverflow.net/questions/63887/non-standard-models-of-finite-set-theory) and [here](http://www.math.nus.edu.sg/%7Ematwtl/papers/finitesetth... | https://mathoverflow.net/users/15293 | Set theory inside arithmetics via the Ackermann yoga | This question reminds me of a magical little-known theorem of Jean Pierre Ressayre that shows that every nonstandard model of $PA$ has a model of $ZF$ as a *submodel* of its Ackermann intepretation, more specifically:
**Theorem.** [Ressayre] Suppose $(M, +, \cdot)$ is a **nonstandard** model of $PA$, and $\in\_{Ack}$... | 16 | https://mathoverflow.net/users/9269 | 67164 | 41,371 |
https://mathoverflow.net/questions/66957 | 3 | Let $X\to \Delta$ be a flat family of complex surfaces with at most a finite number of singularities of simple type, where $\Delta$ is a complex domain in $\mathbb C$.
Here simple type means rational double point.
By a result of Tyurina, it is know that such deformations admit locally simultaneous resolutions afte... | https://mathoverflow.net/users/15600 | Simultaneous resolutions and deformations of simple singularities | I think that the paper Burns-Wahl "Local contributions... " gives some of the examples you are looking for.
(sorry I just wanted to add a comment above, but I am not sure how to do it).
| 2 | https://mathoverflow.net/users/15642 | 67168 | 41,374 |
https://mathoverflow.net/questions/67144 | 4 | Hi,
What is the statement of the Eichler-Shimura relation for Shimura curves? And where
can one find a proof?
Thanks
| https://mathoverflow.net/users/36285 | Eichler-Shimura for Shimura curves | In the general case of Shimura curves over totally real number fields, a nice exposition of the Eichler-Shimura relation is given, for example, in $\S 1.14$ of [this article](http://people.math.jussieu.fr/~nekovar/pu/euler.pdf) of J. Nekovář, where you can find pointers to the relevant literature (in particular, a stan... | 4 | https://mathoverflow.net/users/3642 | 67175 | 41,378 |
https://mathoverflow.net/questions/67169 | 12 | This is my first MO question...hopefully it's not a bad one...
Background: As a stable homotopy theorist, I like to think of complex cobordism $MU$ as a ring spectrum. If I needed to get my hands dirty I could look at the representing spaces or go through the Thom construction of $MU$.
I would like to give a talk a... | https://mathoverflow.net/users/11540 | How is the differential in complex cobordism defined? | For a general audience it is much better to treat $MO$ rather than $MU$, because the complex orientation creates many unpleasant subtleties. (See papers of Buchstaber and Ray for interesting examples where these subtleties make a concrete and computable difference.) Let us say that a geometric chain of dimension n in X... | 18 | https://mathoverflow.net/users/10366 | 67183 | 41,382 |
https://mathoverflow.net/questions/67126 | 2 | Let $\mathbb{K}$ be an arbitrary field with a subfield $\mathbb{F}$ of index 2. Let $a,b\in\mathbb{K}[X]$ be univariate non-vanishing polynomials over $\mathbb{K}$ of degree $\leq 3$ each. **Edit: Due to how this problem arises, one may assume that $a,b$ have no common zeros and at least one has degree exactly 3. Howev... | https://mathoverflow.net/users/8338 | Does "all points rational" imply "constant" for this "cubic" curve over an arbitrary field? | This isn't an answer but I think it's progress. It started off by thinking of restriction of scalars but I've translated it down to a rather more mundane point of view.
Let me call the fields $K$ and $F$ to save some typing.
Let me first deal with the finite field case. My understanding of the question as it curr... | 3 | https://mathoverflow.net/users/1384 | 67184 | 41,383 |
https://mathoverflow.net/questions/67067 | 10 | Let $X$ be a locally compact Hausdorff space, and let $Y\_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}\_x (Y\_t \in \Gamma)$. Let $T\_t$ be the corresponding transition semigroup, i.e. $T\_t f (x) = \int\_X f(y) P(t,x,dy)$. Let $C\_0$ be the set of continuous function... | https://mathoverflow.net/users/4832 | Extending state space to make a process Feller | Yes, it is possible to extend the state space with respect to which $Y$ is a Feller process. Then, $X$ will be a dense open subset of the extension $\hat X$. Furthermore, for any initial distribution of $Y\_0\in\hat X$, then $Y$ will have a continuous modification which necessarily satisfies $Y\_t\not\in\hat X\setminus... | 10 | https://mathoverflow.net/users/1004 | 67186 | 41,384 |
https://mathoverflow.net/questions/67149 | 7 | Suppose we have a cobordism $W$ of manifolds $M\_0$ and $M\_1$ and suppose the inclusion of $M\_0$ into $W$ is a homotopy equivalence. Is the same true for the inclusion of $M\_1$ (ie. is $W$ already an h-cobordism)?
Using Poincare Lefschetz duality one can show that this map induces isomorphisms on homology.
Hence ... | https://mathoverflow.net/users/3969 | Question concerning h-cobordisms | For a counterexample take a non-simply connected homology sphere bounding a contractible manifold and remove the interior of a small ball from the contractible manifold. Such homology spheres exist in abundance.
| 12 | https://mathoverflow.net/users/1822 | 67187 | 41,385 |
https://mathoverflow.net/questions/67171 | 8 | A version of the Hilbert-Mumford criterion states the following: Let $G$ be a linearly reductive group and $V$ a representation of $G$ over a field $k$ (alg. closed, char. zero). Suppose that $y \in \overline{Gx} - Gx$. Then, there is a one-parameter subgroup $\lambda : k^\times \to G$ such that
$$
\lim\_{t\to 0} \lamb... | https://mathoverflow.net/users/4542 | Hilbert-Mumford criterion and closedness | I have a counterexample now, thanks to some notes of Zinovy Reichstein I found. I think the counterexample is paraphrased as follows: Let $V$ be an irreducible representation of $G$ and suppose $x$ does not have a highest weight vector $y$ in its orbit, but $y \in \overline{Gx}$. There will be no way to get to $y$ from... | 9 | https://mathoverflow.net/users/4542 | 67197 | 41,389 |
https://mathoverflow.net/questions/67117 | 25 | This question is so naive that it could have been asked before on this site. If so, I'll delete it.
Among beautiful formula, I like a lot this one:
$$\left(\sum\_{n=1}^Nn\right)^2=\sum\_{n=1}^Nn^3.$$
Is there any other algebraic relation between the polynomials $P\_k$ defined by
$$P\_k(N):=\sum\_{n=1}^Nn^k \qquad?$$
... | https://mathoverflow.net/users/8799 | Relations between sums of powers | <http://en.wikipedia.org/wiki/Faulhaber%27s_formula>
QUOTE:
Faulhaber observed that if $p$ is odd, then
$$1^p + 2^p + 3^p + \cdots + n^p$$
is a polynomial function of
$$a=1+2+3+\cdots+n= \frac{n(n+1)}{2}.$$
END OF QUOTE
QUOTE:
Donald E. Knuth (1993). "Johann Faulhaber and sums of powers". Math. Comp. (A... | 3 | https://mathoverflow.net/users/6316 | 67200 | 41,391 |
https://mathoverflow.net/questions/67195 | 15 |
>
> Let $d$ be an integer. Let $A,B \subseteq \mathbb R^d$ be two sets homeomorphic to an open $d$-ball such that their intersection is again homeomorphic to an open $d$-ball. Does it follow that their union is homeomorphic to an open $d$-ball?
>
>
>
Motivation/Background
---------------------
The motivation c... | https://mathoverflow.net/users/15650 | Homeomorphism type of union of two balls intersecting in a ball. | No to the first question. You can make examples where the "fundamental group at infinity" of $A\cup B$ is nontrivial.
Start with a finite complex $X$ that has trivial homology but nontrivial fundamental group. Embed the suspension $\Sigma X$ in $S^d=\mathbb R^d\cup\infty$. The suspension is contractible and is the u... | 12 | https://mathoverflow.net/users/6666 | 67201 | 41,392 |
https://mathoverflow.net/questions/67199 | 12 | Recently some old notes of mine have gotten me to thinking about the problem of subdividing a triangle into $N$ smaller triangles, all congruent to one another. A little thought shows the following are possible values of $N$:
$\bullet$ If $N$ is a perfect square then any triangle can be subdivided into $N$ such small... | https://mathoverflow.net/users/12301 | Subdivision of triangles into congruent triangles | See <http://www.michaelbeeson.com/research/papers/SevenTriangles.pdf> and <http://www.michaelbeeson.com/research/papers/TriangleTiling1.pdf>
<http://www.michaelbeeson.com/research/papers/TriangleTiling2.pdf>
<http://www.michaelbeeson.com/research/papers/TriangleTiling3.pdf>
Beeson conjectures that the cases you list,... | 12 | https://mathoverflow.net/users/440 | 67204 | 41,395 |
https://mathoverflow.net/questions/67188 | 10 | Let $K$ be a number field, $\chi : C\_K \to \mathbb{C}^\ast$ a Hecke character (that is, a character of the idèle class group), and $L(\chi,s)$ the corresponding Hecke $L$-series. I wish to understand how one may construct a Galois extension $G = Gal(L/K)$ and a complex representation $\rho : G \to \mathbb{C}^\ast$ suc... | https://mathoverflow.net/users/5744 | Recovering Hecke L-series from Artin L-functions | Dear Barinder,
Re. your comment "there cannot be a common generalization of Artin and Hecke $L$-series",
to the contrary, there is such a common generalization, namely the $L$-series of a representation of the global Weil group. These will (conjecturally) have an analytic continuation and functional equation, and the... | 8 | https://mathoverflow.net/users/2874 | 67210 | 41,398 |
https://mathoverflow.net/questions/67192 | 10 | In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is added to the aggregate; then the particle is started from infinity again, and so on.
This begs the question: What does it ... | https://mathoverflow.net/users/3621 | exactly simulating a random walk from infinity | To answer your first question (regarding existence of the limit): I never remember references, but all you have to do here is show that for any two far enough starting points, there is a coupling of the RWs started at them, such that with high probability the two paths hit the aggregate at the same point (for example, ... | 6 | https://mathoverflow.net/users/1061 | 67215 | 41,400 |
https://mathoverflow.net/questions/67207 | 7 | Hi,
Let $D$ be a quaternion algebra over $\mathbf Q$ such that $D\otimes\mathbf R = M\_2(\mathbf R)$.
Let $\pi = \pi\_\infty \otimes\_p \pi\_p$ be an irreducible automorphic representation of $D^\times$.
Supposedly if $\pi\_\infty$ is one-dimensional, then every $\pi\_p$ is one-dimensional, and it should follow f... | https://mathoverflow.net/users/10580 | strong approximation and one-dimensional automorphic representations | Let $G$ be the group of norm one elements in $D^{\times}$. An easy argument
shows that it suffices to prove the claim for $G$ in place of $D^{\times}$.
In other words, I will let $\pi$ be an automorphic rep. of $G$.
Now suppose that $\pi\_{\infty}$
is one-dimensional. This means that in fact $\pi\_{\infty}$ is trivia... | 11 | https://mathoverflow.net/users/2874 | 67221 | 41,403 |
https://mathoverflow.net/questions/67227 | 8 | The construction of the category of finite spectra is easy, but there are different constructions of the whole homotopy category of spectra, all of which leading to the same result up to an equivalence. In his book, *Spectra and the Steenrod Algebra*, North-Holland, Amsterdam, N.Y., 1983, H.R. Margolis conjectured that... | https://mathoverflow.net/users/15541 | Is Margolis's axiomatisation conjecture still alive? | Margolis' conjecture is as follows: Let $\mathcal{S}$ be the stable homotopy category and $\mathcal{T}$ any compactly generated triangulated category. If the subcategories of compact objects are equivalent as triangulated categories $\mathcal{S}^c\simeq\mathcal{T}^c$ then $\mathcal{S}\simeq\mathcal{T}$. This conjecture... | 16 | https://mathoverflow.net/users/12166 | 67230 | 41,405 |
https://mathoverflow.net/questions/67212 | 5 | Let $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties:
1-$P$ preserves GCH and the strong inaccessibility of $\kappa$,
2-$P$ adds a subset of $\kappa$ of size $\kappa$,
3-$P$ is the $< \aleph\_1-$support product of some forcing notions $P\_{\alpha}, \alpha < \kapp... | https://mathoverflow.net/users/11115 | Adding large sets by countable conditions preserving the GCH | Here is my answer to the updated question:
If you allow the forcing $P\_\alpha$ to be trivial, then the Cohen real forcing example still works: add a Cohen real, and then perform $\kappa$ many stages of trivial forcing, with countable support. Overall, this is the same as just adding a Cohen real, since the later sta... | 2 | https://mathoverflow.net/users/1946 | 67236 | 41,408 |
https://mathoverflow.net/questions/67214 | 19 | Let
ZF1 = ZF,
ZFk+1 = ZF + the assumption that ZF1,...,ZFk are consistent,
ZFω = ZF + the assumption that ZFk is consistent for every positive integer k,
... and similarly define ZFα for every computable ordinal α.
Then a commenter on my blog [asked a question](http://www.scottaaronson.com/blog/?p=663#comment... | https://mathoverflow.net/users/2575 | Pi1-sentence independent of ZF, ZF+Con(ZF), ZF+Con(ZF)+Con(ZF+Con(ZF)), etc.? | In 1939, Alan Turing investigated such questions [*Systems of logic based on ordinals*, Proc. London Math. Soc. 45, 161-228]. It turns out that one runs into problems rather quickly due to the fact that the $(\omega+1)$-th such theory is not completely well-defined. Indeed, there are many ordinal notations for $\omega+... | 25 | https://mathoverflow.net/users/2000 | 67237 | 41,409 |
https://mathoverflow.net/questions/67065 | 18 | Suppose we are given a closed oriented 3-manifold.
It is well known that taut foliations are Reebless, and if a Reebless foliation isn't taut then the leaves which don't admit a closed transversal are tori. Furthermore, it is straightforward that in a taut foliation all the closed leaves are homologically non trivial.
... | https://mathoverflow.net/users/12952 | Reebless and taut foliations | For simplicity assume the foliation is transversely oriented (otherwise you can think in terms of the transversely oriented double cover). Therefore each leaf has a canonical orientation. Then a foliation (Reebless or not) is taut if and only if there is no positive linear combination of the oriented torus leaves that ... | 24 | https://mathoverflow.net/users/9062 | 67240 | 41,411 |
https://mathoverflow.net/questions/67225 | 1 | How does the inclusion $\mathbb Z\rightarrow \mathbb Q$ induce a fibration
$K(\mathbb Z,n)\rightarrow K(\mathbb Q,n)$ with fibre $\Omega K(\mathbb Q/\mathbb Z,n)$?
| https://mathoverflow.net/users/15657 | Induced fibration of Eilenberg-MacLane spaces | Probably the most functorial approach is to use the Dold-Kan equivalence
$$F:\{\text{chain complexes}\} \to \{\text{simplicial abelian groups}\}. $$
Let $A\_{\ast}$ denote the chain complex with just $\mathbb{Q}/\mathbb{Z}$ in dimension $n-1$, let $B\_{\ast}$ be the one with a surjective differential from $\mathbb{Q}$... | 3 | https://mathoverflow.net/users/10366 | 67242 | 41,412 |
https://mathoverflow.net/questions/67226 | 3 | Let $G$ be a group, and let $H\leq G$ be a subnormal subgroup. Suppose there exist a cyclic series from $H$ to $G$, that is, a normal series $$H=H\_0\lhd H\_1\lhd\cdots\lhd H\_k= G$$ of subgroups of $G$ such that each factor group $H\_{i+1}/H\_i$ is cyclic. Define the *relative Hirsch number* of the pair $(G,H)$ to be ... | https://mathoverflow.net/users/8103 | Relative Hirsch number | What you are asking is about a generalization of the Hirsch length for polycyclic(-by-finite) groups. Of course, a finitely generated nilpotent group *is* polycyclic, so the special case that mainly interests you is quite classic.
For a polycyclic group (and more generally, for a polycyclic-by-finite group), the Hirs... | 2 | https://mathoverflow.net/users/8338 | 67249 | 41,415 |
https://mathoverflow.net/questions/67248 | 8 | Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we choose a rational representation $\xi : G\to Aut(V)$, we obtain a certain $l$-adic local system $\mathcal L$ on $S$.
Moreo... | https://mathoverflow.net/users/10580 | Is the Galois x Hecke action on cohomology of Shimura varieties semi-simple? | To get a Shimura variety, the reductive group $G$ should satisfy some axioms. In fact,
you should begin not just with $G$, but with a Shimura datum for $G$.
Leaving that aside, the Hecke action will be semi-simple (if we omit Hecke operators at
primes dividing the level); more generally, one could take the limit over... | 8 | https://mathoverflow.net/users/2874 | 67254 | 41,416 |
https://mathoverflow.net/questions/67217 | 3 | Could somebody provide some *simple* examples of Courant algebroids coming from Poisson geometry?
| https://mathoverflow.net/users/14806 | Courant algebroids from Poisson geometry | I think most examples of Courant algebroids come from Poisson geometry. I don't have a particular favorite example: do you have a favorite Poisson manifold? Recall the motivating construction: let $X$ be a (finite-dimensional smooth) manifold, $A \to X$ a (finite-rank smooth) vector bundle, and suppose that both $A$ an... | 6 | https://mathoverflow.net/users/78 | 67256 | 41,417 |
https://mathoverflow.net/questions/67252 | 7 | Recently, I needed to estimate the operator norm of the *tridiagonal operator,* but I am sure answers much more refined than my simple observations must be known.
Let $T$ be the linear operator that maps a square matrix to its tridiagonal part. Thus, the action of $T$ on a matrix $X$ can be defined by the Hadamard pr... | https://mathoverflow.net/users/8430 | Norm of tridiagonal operator | A good answer is given by R. Bhatia: *Pinching, trimming, truncating, and averaging of matrices*. Amer. Math. Monthly **107** (2000), no. 7, 602–608.
If you consider the operator $T\_r$ that retains the diagonals defined by $|i-j|\le r$ (yours is $T\_1$), its norm is accurately bounded by
$$L\_r=\frac{1}{2\pi}\int\_... | 9 | https://mathoverflow.net/users/8799 | 67257 | 41,418 |
https://mathoverflow.net/questions/67050 | 9 | Let $S$ be a scheme and let $N$, $G$ be affine flat group schemes of finite presentation over $S$.
If we assume that $N$ is a closed normal subgroup of $G$, we may form the fppf quotient sheaf $G/N$, which
is a sheaf of groups. By using descent and Artin's representability result for algebraic spaces, it follows that $... | https://mathoverflow.net/users/1084 | Is the category of affine fppf groups closed under normal quotients? | In general, the quotient $G/N$ is not representable. Lemma X.14 of Raynaud's book "Faisceaux amples sur les schémas en groupes et les espaces homogènes" gives a counter-example with $S=\mathbb{A}^2\_k$ (which is regular 2-dimensional), $G=(\mathbb{G}\_{a,S})^2$ and $N\subset G$ étale over $S$.
However if $S$ is local... | 9 | https://mathoverflow.net/users/17988 | 67258 | 41,419 |
https://mathoverflow.net/questions/67259 | 5 | Can countable dense additive subgroups of the reals be well-ordered up to isomorphism by inclusion?
If so, is $\mathbb{Q}$ the smallest (up to isomorphism) countable dense subgroup of the reals, and what is the second smallest (up to isomorphism)?
| https://mathoverflow.net/users/15666 | Countable Dense Sub-Groups of the Reals... | $\{a2^b:a,b\in\mathbb Z\}$ and $\{a3^b:a,b\in\mathbb Z\}$ are both countable dense additive subgroups of the reals, and they are not embeddable in each other (hence $\mathbb Q$ is embeddable in neither).
Also, let $\{p\_k:k\in\mathbb N\}$ be an enumeration of primes, and let $A\_k$ consist of all fractions $a/b$ of i... | 20 | https://mathoverflow.net/users/12705 | 67260 | 41,420 |
https://mathoverflow.net/questions/67271 | 18 | This came up in a question on the xkcd forums. Is it possible to have a nonconstructive metaproof, i.e. a proof that there exists a proof in some formal system which does not construct said proof? Are there any known examples, preferably with some well-known formal system like PA?
Conversely, is it possible to prove ... | https://mathoverflow.net/users/3410 | Are there examples of nonconstructive metaproofs? | In theory, David’s answer is correct. Nevertheless, in practice it is perfectly possible to prove the existence of a proof non-constructively (such as by manipulating models and then appealing to the completeness theorem) where no one has a clue how to actually find the proof.
One example which springs to mind is Jac... | 24 | https://mathoverflow.net/users/12705 | 67273 | 41,426 |
https://mathoverflow.net/questions/67268 | 6 | Suppose that $X$ is a space whose suspension spectrum $\Sigma\_+^\infty(X)$ is dualizable in the stable homotopy category. I believe this is equivalent to saying that $\Sigma\_+^\infty(X)$ is (weakly) homotopy equivalent to a finite cell spectrum. What does this imply about $X$? In particular, does it imply that $X$ is... | https://mathoverflow.net/users/49 | Does a finite suspension spectrum make a space finite? | No. In the stable homotopy category a retract of a finite cell spectrum is again a finite cell spectrum, but in the weak homotopy category of spaces a retract of a finite cell complex is not necessarily a finite cell complex; there is an obstruction in the kernel of $K\_0\mathbb Z[\pi\_1(X)]\to K\_0\mathbb Z$.
EDIT: ... | 11 | https://mathoverflow.net/users/6666 | 67276 | 41,428 |
https://mathoverflow.net/questions/67265 | 9 | It is well known that $4$ general points in $\mathbb{P}^2$ are complete intersection of two conics. On the other hand, if $d \geq 3$, $d^2$ general points are *not* a complete intersection of two curves of degree $d$. More precisely, if $d =3$ there is only one cubic passing through $9$ general points, whereas if $d \g... | https://mathoverflow.net/users/7460 | When is a general projection of $d^2$ points in $\mathbb{P}^3$ a complete intersection? | **Counterexample.** Consider a $Q$ quadric in $\mathbb CP^3$, let $L\_1...,L\_n$, $M\_1,...,M\_n$ be lines on $Q$ so that $L\_i\cap L\_j=\emptyset$, $M\_i\cap M\_j=\emptyset$, while $L\_i$ intersect $M\_j$. Take $n^2$ points $L\_i\cap M\_j$.
**Proof.** For a generic projection $\pi: \mathbb P^3\to \mathbb P^2$ both ... | 14 | https://mathoverflow.net/users/943 | 67278 | 41,429 |
https://mathoverflow.net/questions/67233 | 0 | Looking at semidefinite programs, are there any sufficient conditions for the solvability (i.e. the optimal value can be achieved, that is infimum=minimum)?
Obviously if the problem is unbounded, the optimal value cannot be attained.
Also, if my objective function is continous and the domain is compact, everything is... | https://mathoverflow.net/users/15659 | When can the optimal value of a SDP be achieved? | If the primal problem is feasible and its *dual* problem possesses a *strictly feasible* point, i.e. a point belonging to the (relative) interior of the feasible region, then the primal problem
* is bounded
* attains its optimal value at some point
* has no duality gap with its dual
(note however that under these h... | 0 | https://mathoverflow.net/users/1184 | 67294 | 41,441 |
https://mathoverflow.net/questions/58947 | 11 | Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
**Edit:** To rule out the case where $X$ has infinite genus, perhaps one could add the hypothesis that the topological space $X^{\mathrm{end}}$ (i... | https://mathoverflow.net/users/4721 | Is a non-compact Riemann surface an open subset of a compact one ? | You should probably check the following article:
Migliorini, Luca, "On the compactification of Riemann surfaces".
Here is the Mathscinet review about it:
"In this paper the author studies some questions concerning the compactifications of Riemann
surfaces. It is proved that if $X$ is an open connected Riemann surface th... | 14 | https://mathoverflow.net/users/15673 | 67297 | 41,443 |
https://mathoverflow.net/questions/67295 | 4 | Suppose $L$ is an effective divisor and $H$ is ample (On a smooth 3-fold) such that $L+H$ is nef.
Then show that $L+H$ is big ( $(L+H)^3 > 0$) ?
This was claimed in a paper, without proof. So I assume it should be well-known.
I am not sure if the restriction on dimension is necessary or not.
| https://mathoverflow.net/users/5259 | Question on nef and big divisors | This is indeed true. See for example Lemma 2.60 in Kollar-Mori *Birational Geometry of algebraic varieties*. In particular, it is shown that a Cartier divisor $D$ is big if and only if $mD \sim A + E$ for some ample divisor $A$ and effective divisor $E$. This is also proven in Corollary 2.2.7 in Lazarsfeld's *Positivit... | 5 | https://mathoverflow.net/users/3521 | 67298 | 41,444 |
https://mathoverflow.net/questions/67280 | 3 | Hi!
Given $Z$ a subscheme of a smooth variety $X$ over an algebraically closed field. Is there a way to determine if it is or not reduced by examining the ext sheaves $\mathrm{Ext}^i(O\_Z,\omega\_X)$, where $\omega\_X$ is the dualizing sheaf on $X$? I will be more precise: suppose that I know that there exists $d$ such... | https://mathoverflow.net/users/6949 | Non reduced subschemes and ext sheaves | I just want to clarify, the $Ext$ you are considering are Sheaf-Ext, not global section Ext, right?
Regardless, I think the answer is probably you can't determine that just from vanishing / support. Consider for example $X = \text{Spec } k[x, y]$ and $Z = V(x)$ and $Z'= V(x^2)$. They have the same vanishing behavior ... | 4 | https://mathoverflow.net/users/3521 | 67300 | 41,446 |
https://mathoverflow.net/questions/67251 | 6 | We have convex sets $C\_1=Conv(yy^{T}|y^{T}y=a,y\in R^{M})$ and $C\_2=Conv(yy^{T}|y^{T}y=a,y\in R\_{\geq 0}^{M})$. Clearly $C\_2\subset C\_1$. Does there exist a PSD matrix $A$ having $tr(A)=a,A(i,j)\geq 0$ $\forall i,j$ and $A\in C\_1\setminus C\_2$?
| https://mathoverflow.net/users/39663 | PSD matrix with non-negative entries | There is such an $A$ if and only if $M\geq 5$.
To see this, first note that the condition that $A$ be a convex combination of terms $yy^T$ each with trace $a$ is irrelevant. As long as $A$ is positive semidefinite (and symmetric), it can be written as a convex combination of terms $yy^T$. If $A$ has trace $a$ then li... | 4 | https://mathoverflow.net/users/5963 | 67301 | 41,447 |
https://mathoverflow.net/questions/67290 | 6 | I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach space. I think that everyone who tries to study "classical" operator spaces like $\mathcal{K}$, Schatten $p$-class operato... | https://mathoverflow.net/users/15129 | Space of compact operators | It is easy to see that whenever a space has an unconditional basis then the space of diagonal operators of the basis is equivalent to $\ell\_\infty$. If $c\_0$ embeds in $K(X,Y)$ then $K(X,Y)$ is not complemented in $B(X,Y)$. One reference for this is: M. FEDER. On subspaces of spaces with an unconditional basis and sp... | 9 | https://mathoverflow.net/users/15388 | 67303 | 41,449 |
https://mathoverflow.net/questions/67306 | 3 | Reading a paper on hamiltonian mechanics, in a section on classical examples of complete integrability, it is examined the geodesic flow of a triaxial ellipsoid.
Before separating the variables in the Hamilton-Jacobi equation, the original approach followed by Jacobi himself, it is stated that the phase space of this... | https://mathoverflow.net/users/12617 | what are the killing vector fields on a triaxial ellipsoid? | This follows from a rigidity theorem for convex surfaces. According to the Wikipedia page on [Cauchy rigidity](http://en.wikipedia.org/wiki/Cauchy%2527s_theorem_%2528geometry%2529), this was proved by Cohn-Vossen for smooth convex surfaces in $R^3$. So two convex surfaces which are intrinsically isometric are related b... | 7 | https://mathoverflow.net/users/1345 | 67307 | 41,450 |
https://mathoverflow.net/questions/67310 | 2 | Let V be the universe (the class of all sets), let W(0)=V, W(1) be the class of all singletons whose unique member element is a member set of W(0), and for n>0 let W(n+1) be the class of all singletons whose unique member element is a member set of W(n).
Let, for every n>=0 S(n)=W(n+1)/W(n) be the class that is the dif... | https://mathoverflow.net/users/30395 | Finitely nested Singletons and the axiom of Regularity | I assume that you meant to write $S(n)=W(n)-W(n+1)$, rather than what you have written, since inductively one can show $W(n+1)\subseteq W(n)$. With this understanding, all the $S(n)$ are disjoint, and the question is whether every set eventually falls out, or whether there can be a set in every $W(n)$.
In ZFC, there... | 6 | https://mathoverflow.net/users/1946 | 67314 | 41,453 |
https://mathoverflow.net/questions/67318 | 12 | I believe this may be a standard algebraic topology problem, so I apologize in advance if this belongs in stackexchange (it's not a homework problem, however, and came about in a research context). I've got a continuous map $f$ from the $n$-simplex to itself, such that the image of every strict sub-simplex is itself. S... | https://mathoverflow.net/users/13363 | Map from simplex to itself that preserves sub-simplices | Given such a map $f:\Delta\_n\to\Delta\_n$, put $f\_t(x)=(1-t)x+t f(x)$. This gives a homotopy between $f$ and the identity, and each map $f\_t$ also sends every subsimplex to itself. In particular, each $f\_t$ preserves $\partial(\Delta\_n)$ and so induces a self-map $\overline{f}\_t$ of the space $\Delta\_n/\partial(... | 19 | https://mathoverflow.net/users/10366 | 67319 | 41,456 |
https://mathoverflow.net/questions/67305 | 5 | Let $\Delta\_{+}$ be the sub-category of the simplex category $\Delta$ containing only injective functions, and take $M$ to be a nice model category. I'll write $i \colon \Delta\_{+} \hookrightarrow \Delta$ for the inclusion.
Now assume we have a semi-simplicial diagram $X \colon \Delta\_{+}^{\text{op}} \to M$. We ca... | https://mathoverflow.net/users/473 | Does adding degeneracies to a semi-simplicial diagram change the homotopy colimit? | As I understand it, homotopy left Kan extension along any functor $i:\mathcal C\to\mathcal D$ preserves hocolim.
Left Kan extension $i\_!$ is left adjoint to restriction $i^\star$, i.e. composition with $i$. (Of course, if $i$ is not fully faithful, "extension" is a bit of a misnomer: restriction composed with "exte... | 10 | https://mathoverflow.net/users/6666 | 67320 | 41,457 |
https://mathoverflow.net/questions/67324 | 9 | I'm reading Mike Hopkins' [COCTALOS](http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf) notes and having trouble with some pretty basic statements about (naive) spectra. Basically I'm nervous doing homological algebra with them, although in these problems I'm having I couldn't imagine the proofs bei... | https://mathoverflow.net/users/303 | homological algebra with spectra | Suppose that the inclusion $\eta\wedge 1:I\to E\wedge I$ admits a retraction, say $r$. Consider an $E$-monomorphism $f:A\to B$ and an arbitrary map $g:A\to I$. As $f$ is an $E$-monomorphism we can choose a retraction $s:E\wedge B\to E\wedge A$, and then the composite
$$ h = (B \xrightarrow{\eta\wedge 1} E\wedge B \xri... | 5 | https://mathoverflow.net/users/10366 | 67327 | 41,461 |
https://mathoverflow.net/questions/67343 | 6 | Let $A$ be a commutative algebra (over the complex numbers, with a unit) and let $M$ be a finitely generated projective $A$-module, and let $m\_1,\ldots,m\_n$ be a set of generators of $M$. The Dual Basis Theorem states that there exists $m\_1^\ast,\ldots,m\_n^\ast\in M^\ast$ such that $x=\sum m\_i^\ast(x)m\_i$ for all... | https://mathoverflow.net/users/15488 | Trace of the identity map in a projective module | More abstractly: Let $A$ be a commutative unital ring and let $M$ be a f.g. projective $A$-module. The rank $rk\_P(M)$ of $M$ at a prime ideal may be defined as the vector space dimension of $K\otimes\_A M$ where $K$ is the residue field. Because projective modules of local rings are free, this is a locally constant fu... | 9 | https://mathoverflow.net/users/6666 | 67345 | 41,467 |
https://mathoverflow.net/questions/67312 | 1 | Let V be the universe of sets (the class of all sets). Let U(0)=V, U(1)=V\*V, the class that is cartesian product of the class V=U(0) with V, and for n>=1, let U(n+1)=U(n)\*U(0);
For every natural integer n, let T(n)=U(n+1)/U(n) be the class that is the difference of the class U(n+1) and of the class U(n).
We are inter... | https://mathoverflow.net/users/30395 | Finite T-uples and the axiom of Regularity | Question 3 also has a negative answer (Joel Hamkins has already answered the first two questions).
The model $V(a,b,c)$ described in detail in my answer to an analogous [question](https://mathoverflow.net/questions/67310/finitely-nested-singletons-and-the-axiom-of-regularity) works here as well since *$V(a,b,c)$ sati... | 4 | https://mathoverflow.net/users/9269 | 67353 | 41,471 |
https://mathoverflow.net/questions/67355 | 4 | What is a simple way to prove that for any compact two-dimensional surface $S$ and an element $g$ in $\mathbb \pi\_1(S)$ there exists a finite index normal subgroup $\Gamma\subset \pi\_1(S)$ such that $g\notin \Gamma$?
In fact, who was first to prove this statement?
| https://mathoverflow.net/users/13441 | Residual finiteness of fundamental groups of surfaces. | See [this text.](http://www.math.umbc.edu/~campbell/CombGpThy/RF_Thesis/3_Knot_Manifold_Groups.html) The proof is just a few lines.
| 11 | https://mathoverflow.net/users/nan | 67357 | 41,474 |
https://mathoverflow.net/questions/67329 | 7 | In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts.
Let $B=\sqcup\_{n\in\Bbb N} BS\_n$, which I'd like to think of as the [categorified](https://mathoverflow.net/questions/4841) version of the natural numbers $\Bbb N$. There is an ... | https://mathoverflow.net/users/10819 | categorifying induction in homotopy type theory | The first reason you give is sufficient to answer your question: any interpretation of `nat` (and any other type with decidable equality) must have contractible components. Let me try to unpack the proof:
The proof of `isasetifdeceq` goes as follows: Fixing $x:X$, we must show that $\text{Paths}(x,x)$ is contractible... | 7 | https://mathoverflow.net/users/2004 | 67361 | 41,475 |
https://mathoverflow.net/questions/67359 | 3 | Given two multisets $A$ and $B$ of the same finite cardinality $n$, how many ways are there of pairing the two sets together?
If both sets consist of distinct elements, the answer is $n!$: there are $n$ ways to pair the first element of $A$ with something from $B$, $n-1$ for the second element, etc. If one of the set... | https://mathoverflow.net/users/2056 | The number of pairings between multisets | If the multiplicities of the elements of the first multiset are $a\_1,a\_2,\dots$ and of the second $b\_1,b\_2,\dots$, then you are asking for the number of matrices $A=(A\_{ij})\_{i,j\geq 1}$ of nonnegative integers with row-sum vector $(a\_1,a\_2,\dots)$ and column-sum vector $(b\_1,b\_2,\dots)$. These are very well-... | 12 | https://mathoverflow.net/users/2807 | 67365 | 41,478 |
https://mathoverflow.net/questions/67366 | 3 | Assume AC. Suppose $X$ is a subset of the irrationals (Baire Space) for which neither player has a winning strategy (i.e. the game $G(\omega, X)$ is not determined). Is $X$ non-measurable in the Lebesgue sense as a subset of $\mathbb{R}$?
| https://mathoverflow.net/users/15666 | Non-measurable sets and Determinacy... | (My argument is somewhat easier if you consider games where
the players play $0$s and $1$s, so that the payoff set is
in Cantor space $2^\omega$, and we use the usual
coin-flipping probability measure; but an essentially
similar idea works in Baire space.)
For any game with payoff set $A$, where player I wins if
the ... | 6 | https://mathoverflow.net/users/1946 | 67368 | 41,479 |
https://mathoverflow.net/questions/67363 | 24 | **What is the automorphisms group of the weighted projective space $\mathbb{P}(a\_{0},...,a\_{n})$ ?**
Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any automorphism has to fix the two singular points. Consider a smooth point $p\in\mathbb{P}(2,3,4)$. **What is the sub... | https://mathoverflow.net/users/14514 | Automorphisms of a weighted projective space | The automorphism group is the quotient of the automorphism group of the corresponding graded algebra by 1-dimensional torus acting by rescaling. In the particular case of $P(2,3,4)$ the graded algebra is $A = k[x\_2,x\_3,x\_4]$ with $\deg x\_i = i$. Note that any automorphism should take $x\_2 \to a x\_2$, $x\_3 \to b ... | 19 | https://mathoverflow.net/users/4428 | 67370 | 41,481 |
https://mathoverflow.net/questions/67373 | 2 | Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that
$$
\|f(x)-f(y)\| \;\geq\; \|x-y\|
$$
for all $x,y \in S$. Does it follow that $\mu(S) \leq \mu(T)$?
| https://mathoverflow.net/users/6514 | Expanding Measurable Sets | It follows from two observations:
* For Hausdorff meeasure your statement follows from the definition.
* Hausdorff measure = Lebesgue measure (up to constant).
| 7 | https://mathoverflow.net/users/1441 | 67376 | 41,484 |
https://mathoverflow.net/questions/67371 | 5 | The [Łos-Tarski preservation theorem](http://en.wikipedia.org/wiki/%C5%81o%C5%9B%E2%80%93Tarski_preservation_theorem) states that a set of formulas $F$ of first-order language $L$ is preserved under substructures for models of theory $T$ in $L$ precisely when $F$ is equivalent modulo $T$ to a set of universal formulas ... | https://mathoverflow.net/users/7252 | Higher-order preservation theorems? | There are a number of preservation results concerning the infinitary logic $L\_{\omega\_{1},\omega}$, as well as its so-called *admissible fragments*. Such results include an analogue of the one you mentioned about existential sentences.
Here are some basic sources in this direction:
H.J. Keisler, **Model theory fo... | 7 | https://mathoverflow.net/users/9269 | 67380 | 41,486 |
https://mathoverflow.net/questions/67383 | 2 | It is well-known that the braid group $B\_{n}$ injects into the group of automorphisms of the free group $F\_{n}$. However, there is certainly a kernel when mapping to the outer automorphism group $Out(F\_{n})$. Namely, the kernel contains the generator of the center of $B\_{n}$. Could someone please explain or give a ... | https://mathoverflow.net/users/6254 | Center of the braid group and outer automorphisms of the free group | Assuming we're talking about the same map and there's just a $n \longmapsto n+1$ mix-up, this question reduces to studying $Inn(F\_n)$ intersected with the image of $B\_n \to Aut(F\_n)$.
An inner automorphism of $F\_n$ is a conjugation automorphism, so we're looking at braids that act on $F\_n$ by conjugation. Let t... | 5 | https://mathoverflow.net/users/1465 | 67385 | 41,488 |
https://mathoverflow.net/questions/67387 | 26 | Let $X$ and $Y$ be varieties. Let $E$ be a locally free sheaf over $X$. Let $f: X \to Y$. Is there some nice criteria which ensures that $f\_\ast E$ is still locally free? Sorry, if this is a very standard question.
| https://mathoverflow.net/users/15692 | When is the pushforward of a vector bundle still a vector bundle? | Under reasonable hypotheses on $X$, $Y$ and $f$, the answer is that $\dim H^0(X\_y, \, E\_y)$ is a *constant* function implies $f\_\*E$ is locally free, where
$$X\_y:=f^{-1}(y), \quad E\_y:=E|\_{X\_y}.$$
More precisely, there is the following result, whose proof can be found in [Mumford, *Abelian Varieties*, Chapte... | 29 | https://mathoverflow.net/users/7460 | 67394 | 41,491 |
https://mathoverflow.net/questions/67342 | 2 | I'm currently trying to understand the proof of the (elementary) JSJ-Decomposition in [Allen Hatcher's notes on 3-Manifolds](http://www.math.cornell.edu/%7Ehatcher/3M/3Mdownloads.html).
Specifically, I have trouble understanding the application of proposition 1.7 to corollary 1.8.
Proposition 1.7 states:
>
> Fo... | https://mathoverflow.net/users/13554 | JSJ-Decomposition - proof of finiteness in Hatcher's Notes | Since we can only do JsJ-decomposition for toroidal manifold, as you say, we can assume that manifold M is toroidal. cutting M along the essential torus, the induced manifold or manifolds
are both not the surface I-bundle and possible toroidal. We can do Jsj-decomposition again and again until that each component is a... | 2 | https://mathoverflow.net/users/15484 | 67401 | 41,497 |
https://mathoverflow.net/questions/67407 | 1 | Hey,
Is it possible to define a map between two categories which preserves all products and binary equalizers and yet is not a functor, ie it does not satisfy one or more axioms of a functor? Further, if you replace the functor with maps between categories that preserve all products and binary equalizers, what do the... | https://mathoverflow.net/users/10007 | continuous maps between categories that are not functors | It does not make much sense to talk about non-functors $F$ which preserve products because there is no canonical morphism $F(x \times y) \to F(x) \times F(y)$. Even if we say that we choose one, we cannot formulate compatibility properties by varying $x,y$. So this will be just some random isomorphism, perhaps somethin... | 6 | https://mathoverflow.net/users/2841 | 67408 | 41,501 |
https://mathoverflow.net/questions/67382 | 16 | The non-zero elements of the minimal prime ideals of a noetherian commutative ring are zero-divisors.
The proof of this fact I know of uses primary decomposition. Are there alternative (e.g. more direct) proofs ?
| https://mathoverflow.net/users/10194 | Minimal primes and zero-divisors | Let $A$ be your ring and $X=\mathrm{Spec} A$. The minimal primes of $A$ correspond to the irreducible components of $X$. An element of $f\in A$ induces a function $\widehat f:X\to \coprod\_{\mathfrak p\in \mathrm{Spec} A}\kappa(\mathfrak p)$ and this function vanishes everywhere if and only if $f\in\mathfrak p$ for all... | 14 | https://mathoverflow.net/users/10076 | 67413 | 41,503 |
https://mathoverflow.net/questions/67409 | 2 | The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square.
Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D\_r \supseteq I\_r$ be a decomposition group and inertia group at $r$. Let $E/\mathbb{Q}$ be an elliptic curve, and for a prime $\ell \neq r$, consider th... | https://mathoverflow.net/users/2615 | How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction? | The factor $D\_r$ is easy to compute (much easier than $\mathcal{D}\_r$). Basically, you just need to find the eigenvalues $\lambda\_i$ of Frobenius on $H^1\_\ell(E)^{I\_r}$ (i.e. the reciprocal roots of the local L-factor of E itself), and then the eigenvalues of Frobenius on the symmetric square of that are the pairw... | 2 | https://mathoverflow.net/users/2481 | 67415 | 41,505 |
https://mathoverflow.net/questions/67390 | 2 | Hi,
Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?
Thanks
| https://mathoverflow.net/users/36285 | nth-powers and degree n polynomials with coefficients in field extensions | If $P$ is monic, the answer is Yes (in any characteristic, if the extension is separable), because a field automorphism over $F$ fixes $P^n$ and $(P^{\sigma})^n=P^n$ implies $P^{\sigma} = P$ for monic polynomials. In general, if $c$ is the leading coefficient of $P$, then $c^n \in F$ and
$c^{-1}P\in F[X]$. Thus the cou... | 3 | https://mathoverflow.net/users/10266 | 67420 | 41,506 |
https://mathoverflow.net/questions/67417 | 3 | Suppose that I have a morphism of schemes of finite type over $\mathbb{C}$, which is a bijection on closed point (but in general I don't know if there exists an inverse for it).
I would like to know if there exist sufficient conditions on the source or the target such that the induced morphism of Hodge structures is ... | https://mathoverflow.net/users/11060 | When does a bijective morphism of schemes induce an isomorphism of Hodge structures? | The induced morphism of Hodge structures for any map of varieties will be an isomorphism if and only if the induced map on cohomology (forgetting the Hodge structure) is an isomorphism. This follows from the fact that morphisms of mixed Hode structures are strict with repsect to both the Hodge and weight filtration; se... | 14 | https://mathoverflow.net/users/519 | 67423 | 41,508 |
https://mathoverflow.net/questions/67434 | 1 | Let $f: \mathbb{R}^n \to L^2(\mathbb{R}^d) $ be a Bochner-integrable function (all measures are the Lebesgue measure). Does then $ \int\_{\mathbb{R}^n} f(x) d\lambda^n (y) = \int\_{\mathbb{R}^n} f(x)(y) d\lambda^n $ hold for $\lambda^d$-almost all $y \in \mathbb{R}^d$? I.e. can one compute such Bochner integrals just b... | https://mathoverflow.net/users/13338 | Computing Bochner integrals with values in L^p-spaces by Lebesgue integrals? | Answer: YES and NO.
YES: In any practical situation you are likely to meet, your formula is correct. You would prove it using Fubini's Theorem, pairing your two sides with an arbitrary $h \in L^2(\mathbb R^d)$ and getting the same answer on both sides. The catch is, you have to be able to apply Fubini.
NO: As sta... | 6 | https://mathoverflow.net/users/454 | 67442 | 41,515 |
https://mathoverflow.net/questions/67419 | 1 | I would like to know if there are any standard techniques (that I don't know about) to solve the following problem.
Suppose we have $n$ variables, $\mathbf{q} = (a\_1, a\_2, \ldots, a\_n)$, but not all of them are independent. For example, the values could be determined by only a single variable, e.g. $(x, x^2, x^3)$... | https://mathoverflow.net/users/8776 | Finding number of independent variables in "statistical" dataset | There should be quite an extensive literature on this type of problem. A quick Google search turned up these papers:
<http://www.cs.bu.edu/techreports/pdf/2011-012-intrinsic-dimension-clustering.pdf>,
<http://www.princeton.edu/~wbialek/our_papers/chigirev+bialek_04.pdf>
This, and further references given there, shoul... | 2 | https://mathoverflow.net/users/12120 | 67447 | 41,518 |
https://mathoverflow.net/questions/67448 | 13 | So, I've been trying to understand what exactly an anomaly is, and how they arise in physics. Apparently an anomalous theory is some theory whose action is given by a section of some bundle (rather than a function). Hence, only if the bundle is topologically trivial, thus allowing one to write the action as a function,... | https://mathoverflow.net/users/13132 | Nice example of a topologically trivial bundle with nontrivial connection | I think that what you say is not complitely correct: the local anomaly is the curvature of the connection, which can be non-trivial wether the bundle is trivial or not. The global anomaly is the holonomy of the connection, which can be non-vanishing even for flat connections. In fact, considering the case of a line bun... | 10 | https://mathoverflow.net/users/10758 | 67452 | 41,520 |
https://mathoverflow.net/questions/67455 | 15 | Let $f\colon\thinspace E\to B$ be a Serre fibration whose fibre $F$ is $k-1$-connected, $k\geq 1$. Assume $B$ is a connected CW complex. Then the primary obstruction to the existence of a cross section of $f$ is defined; it is a cohomology class $$\mathfrak{o}(f)\in H^{k+1}(B;\tilde{H}\_{k}(F)).$$
Here the coefficients... | https://mathoverflow.net/users/8103 | Where does the primary obstruction of a fibration show up in its spectral sequence? | At least in the case $\pi\_1(B)=0$, $\mathfrak{o}(f)$ is just the first non-trivial differential, $d\_k$ in disguise (let's work over some field, for simplicity; then $d\_k\in\operatorname{Hom}(H^k(F),H^{k+1}(B))\cong H^{k+1}(B)\otimes H\_k(F)\ni \mathfrak o(f)$).
Reference (well, kind of: it doesn't even give precis... | 7 | https://mathoverflow.net/users/1556 | 67457 | 41,522 |
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