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https://mathoverflow.net/questions/82239 | 2 | For $y\in \mathbb{R}$ and $P \subset \mathbb{R}$, consider the distance $d(y,P) = \inf\_{x\in P} |x-y|$.
Given arbitrary $\alpha, \beta, \gamma \in \mathbb{R}$, I am interested to know how to find the smallest $k \in \mathbb{N}$ s.t. $d(\alpha + \beta k,\mathbb{Z}) < \gamma$.
There are some special cases where the... | https://mathoverflow.net/users/1680 | Elementary question about 'Progressions avoiding sets' | I like this question and the answer, which doesn't seem to appear on either of the pages referenced above, comes from what is called the *Ostrowski expansion* of a real number (some references use other names for this expansion). This is an analogue of the base $p$ expansion which does for the $d(\cdot, \mathbb{Z})$ di... | 8 | https://mathoverflow.net/users/19368 | 82262 | 49,238 |
https://mathoverflow.net/questions/82261 | 4 | Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.
1. Can I say something about the fibers over non-closed points?
2. Is it true that $X$ is normal?
In the local setting, a positive answer to 2 can be f... | https://mathoverflow.net/users/19634 | Flat family of normal schemes over a normal base | The fibers are normal. This follows (without the normality assumption on $Y$) from EGA IV (12.1.6) which says that the set those $x\in X$ where the fiber is normal is open, hence if its complement were nonempty, it would contain a closed point.
Hence, by your remark (or by EGA IV (6.8.3)), $X$ is normal.
| 9 | https://mathoverflow.net/users/7666 | 82264 | 49,239 |
https://mathoverflow.net/questions/82248 | 1 | Let $v\in \mathbb{C}^n$ be an $n$ dimensional complex vector. Define the non-standard bilinear form $\left< u,v \right> = u^T v$ (the usual inner product except without the conjugation). What are the properties of the self orthogonal vectors, those such that $\left < v,v \right>=0$? What are the properties of this spac... | https://mathoverflow.net/users/1074 | Characterizing the set of self-orthogonal complex vectors | The most natural way to view your set of vectors is in the setting of projective geometry; the vector space $K^n$ (where $K$ is now an arbitrary field, so $K = \mathbb{C}$ for you) can be seen as a projective space $\mathbb{P}^{n-1}(K)$ of dimension $n-1$. (The "points" of the projective space $\mathbb{P}^{n-1}(K)$ cor... | 3 | https://mathoverflow.net/users/12858 | 82268 | 49,241 |
https://mathoverflow.net/questions/82260 | 6 | I am a total non-expert, so the answer to this question may be obvious, but here goes.
In Chevalley's formulation of CFT we get Artin maps $J\_k \rightarrow Gal(L/k)$, where $J\_k$ is the group of all ideles of $k$. However, we know there is a nice subgroup $J\_k^1$ of the ideles obtained by taking only those satisfy... | https://mathoverflow.net/users/8080 | Class field theory using only ideles of norm 1 | I am no expert either, but here is what I think. If $k$ is a global field of characteristic zero (i.e. an algebraic number field), then one can work with $J\_k^1$ instead of $J\_k$ without losing (or changing) anything. This is because the kernel of the Artin map contains a subgroup $N$ of $J\_k$ isomorphic to the mult... | 9 | https://mathoverflow.net/users/11919 | 82282 | 49,247 |
https://mathoverflow.net/questions/82227 | 7 | **Theorem.** Let V be a $C^\infty$ function on a riemannian manifold $M$ and $p$ be a nondegenerate local minimum with $V(p)=0$. Then there is a unique positive function $\varphi \in C^\infty(U)$ such that $\varphi$ solves the eikonal equation
$$\|\mathrm{grad} \varphi \|^2 = V.$$
Here, $U$ is some open neighborhood of... | https://mathoverflow.net/users/16702 | Solutions to the eikonal equation | Note: I have realized that, using the Stable Manifold Theorem, one can prove the smoothness of the solution $\phi$ that I describe below. Thus, I am modifying my answer to incorporate that.
Local existence and uniqueness of a smooth solution near $p$ satisfying $\phi(p)=0$ and $\phi\ge0$ near $p$ follows from an app... | 10 | https://mathoverflow.net/users/13972 | 82284 | 49,248 |
https://mathoverflow.net/questions/82286 | 2 | Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:
$$n-i-1=(d-1)\sum\_{l=1}^{n-i-1}\frac{\binom{n-i-1}{l}}{\binom{n-i+d-3}{l}}$$
| https://mathoverflow.net/users/19642 | Rational Binomial Identity | The canonical reference for this sort of thing is Petkovsek and Zeilberger's book "A=B". Maple (almost certainly) uses the Zeilberger-Wilf algorithm for hypergeometric summation (which really goes back to Bill Gosper). You can also read the Wilf-Zeilberger paper (Inventiones, around 1990).
| 3 | https://mathoverflow.net/users/11142 | 82298 | 49,252 |
https://mathoverflow.net/questions/82296 | 11 | Does anyone know of an explicit effective lower bound for $|L(1,\chi)|$, where $\chi$ is an odd complex (primitive) Dirichlet character?
I know of Landau's paper Uber Dirichletsche Reihen mit komplexen Charakteren, where he bounds $$ |L(1,\chi)|>\frac{1}{c \log(q)},$$
where $q$ is the conductor of $\chi$, but the c... | https://mathoverflow.net/users/19646 | explicit lower bounds on $|L(1,\chi)|$ | This is discussed on page 47 of Narkiewicz's new book (Rational Number Theory in the 20th Century); see
<http://books.google.ca/books?id=3SWNZaDM6iMC&lpg=PP1&dq=rational%20number%20theory%20in%20the&pg=PA47#v=onepage&q&f=false>
Reference [4268] is to
Metsankyla, T.: Estimations for L-functions and the class numb... | 10 | https://mathoverflow.net/users/16510 | 82299 | 49,253 |
https://mathoverflow.net/questions/82291 | 2 | I have a special type of companion matrix, where the "special" part is that each element in the matrix are matrices. For instance, the diagonal with "1":s is instead a diagonal with identity matrices, and so on...
My question is if anyone knows of a solid algorithm for finding the eigen-decomposition of such a matrix?... | https://mathoverflow.net/users/19600 | eigen-decomposition of a special companion matrix | You can get some savings with respect to the naive $O(n^3d^3)$ by using the semiseparable structure, but I don't think you can get anything faster than $O(n^3d^2)$ (stable) or $O(n^3d\log d)$ (maybe unstable). Algorithms for semiseparable matrices aren't exactly easy to start working with unless you work in numerical l... | 1 | https://mathoverflow.net/users/1898 | 82304 | 49,255 |
https://mathoverflow.net/questions/82196 | 2 | Let $F$ be a local field and $G = GL(n,F)$. Let $f$ be an element $C\_c^\infty(G)$.
Let $\gamma$ be an elliptic element of $G$ with irreducible characteristic polynomial.
What are strategies to compute
$$ \int\limits\_{G\_\gamma \backslash G} \phi(g^{-1}\gamma g) \mathrm{d} g?$$
Due to a comment of Paul Broussous... | https://mathoverflow.net/users/10400 | Elliptic orbital integral | You have introductions to the building of ${\rm GL}(n)$ in Brown's book "Buildings", or in Paul Garrett's book. The extended building is just the product of the non extended building $X$ with the real line $\mathbb R$, with the action
$$
g.(x,r)=(\ g.x \ , \ r+val\_F (det(g)) )
$$
It is the geometric realization of a ... | 1 | https://mathoverflow.net/users/4767 | 82308 | 49,259 |
https://mathoverflow.net/questions/82040 | 2 | Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q(R)$, the fraction field of $R$, and set $K'=Q(\hat{R})$. Must the degree of the field extension $K'/K$ be infinite? The answer is "yes" for ... | https://mathoverflow.net/users/7103 | Quotient field extension for an incomplete DVR | Nagata's example (E3.3) shows that you can have $[K':K]=p<\infty$. Thanks to Bruce Olberding for the pointer.
| 1 | https://mathoverflow.net/users/7103 | 82313 | 49,262 |
https://mathoverflow.net/questions/82312 | 5 | The question is mostly in the title:
>
> What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?
>
>
>
Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary rotations as i... | https://mathoverflow.net/users/17193 | What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space? | This is the "piano movers problem", also known as the motion planning problem, which has an enormous literature. Check out
<http://en.wikipedia.org/wiki/Motion_planning>
| 7 | https://mathoverflow.net/users/11142 | 82315 | 49,263 |
https://mathoverflow.net/questions/82292 | 3 | Recently I have been trying to understand Deigne-Lustzig induction in the case of $G = \text{Sl}(2,\mathbb{F}\_p).$
In this case the appropriate Deligne Lustzig variety is given by $X:xy^q-y^qx = 1,$ aka the Drinfeld curve. The action of $G$ on affine space fixes $X$ and in addition commutes with the action of the $q... | https://mathoverflow.net/users/12693 | Action of Non-Split Torus in Deligne-Lustzig induction | It would be helpful to know what sources you are consulting, since the rank 1 case is only a warm-up to the general Deligne-Lusztig theory. Their 1976 *Annals* paper (followed by a considerable amount of detailed work by Lusztig and others)
treats arbitrary finite groups of Lie type and their complex characters in a no... | 5 | https://mathoverflow.net/users/4231 | 82316 | 49,264 |
https://mathoverflow.net/questions/82311 | 4 | I have a question about generating a certain set of symmetric polynomials. I believe that what I'm looking for is known result, but I'm not 100% sure.
Suppose that I have two sets of indeterminates $X\_1,\ldots,X\_n$ and $Y\_1, \ldots, Y\_n$, of the same (finite) cardinality. Now let $e\_i(A,B,C,\ldots)$ denote the ... | https://mathoverflow.net/users/14157 | Constructing the sum of two pairs of symmetric polynomials | What you want is not possible. Suppose $n>3$, and consider making $A\_2=\sum\_{i < j}(X\_iX\_j+Y\_iY\_j)$. All polynomials available are homogeneous in the total degree, and the degree of the $C\_i$ is too high to be of any use, so you'll have to make do with the $B\_i$. But the only way to get a monomial $X\_iX\_j$ is... | 3 | https://mathoverflow.net/users/19077 | 82322 | 49,267 |
https://mathoverflow.net/questions/82232 | 2 | Let $\pi:Y\to X$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$.
**Question.** Does there exist a finite field extension $L/K$ and a regular model $\mathcal{X}/O\_L$ for $X\_L/L$ such that the minimal resolution of singularities of the normalization of $\mathcal{X}$ ... | https://mathoverflow.net/users/4333 | Does each finite morphism of curves have a model whose minimal resolution is semi-stable | If $X$ has no (potentially) good reduction, then the answer to your question is no. More precisely, there always exists a finite cover $Y\to X$ such that for any finite extension $L/K$, no regular semi-stable model of $Y\_L$ dominates a regular semi-stable model of $X\_L$.
Suppose we are given a finite morphism of s... | 3 | https://mathoverflow.net/users/3485 | 82328 | 49,270 |
https://mathoverflow.net/questions/82288 | 5 | Let $(x\_1 \ldots ,x\_n) \in \mathbb{R}^n$ and $f\_i = \Pi\_{j=1, j \neq i }^n ( x\_i - x\_j )$
I'm trying to evaluate $(f\_1, \ldots, f\_n)$. A trivial algorithm runs in $\mathcal{O}(n^2)$ but given the very specific form of the problem, there's got to be something faster. Maybe I've overlooked something simple, may... | https://mathoverflow.net/users/8737 | Fast (subquadratic) evaluation of a class of N degree polynomials over N points | The following may be of some help, if you haven't thought of it already. Let $V$ be the Vandermonde matrix with $(i,j)$th entry $x^{i-1}\_{j}$, $i,j=1,\ldots,n$. Its inverse $W$ has $(i,1)$th entry
$$
(-1)^{n} \frac{x\_1 \ldots x\_{i-1} x\_{i+1} … x\_n}{f\_i}.
$$
Hence, to find $f\_1,\ldots,f\_n$ we need to solve $V \a... | 2 | https://mathoverflow.net/users/19011 | 82342 | 49,277 |
https://mathoverflow.net/questions/82277 | 7 | Does anyone know of an easy proof of Shannon-McMillan-Brieman theorem?
Thanks
| https://mathoverflow.net/users/19639 | Shannon-McMillan-Breiman theorem | In addition to Igor's [answer](https://mathoverflow.net/a/82278), there's also:
D. Ornstein and B. Weiss, "[The Shannon–McMillan–Breiman theorem for a class of amenable groups](https://doi.org/10.1007/BF02763171)", *Israel J. Math.* **44** (1983), 53–60. [Zbl 0516.28020](https://zbmath.org/0516.28020)
| 3 | https://mathoverflow.net/users/5701 | 82344 | 49,279 |
https://mathoverflow.net/questions/82271 | 1 | Let $(R,\mathfrak{m})$ be a complete local ring, $a\_{\lambda}$ be a decreasing net of ideals in $R$, indexed by a directed set. Consider the completion under $a\_{\lambda}$-topology $A=\underleftarrow{\lim} R/\mathfrak{a}\_{\lambda}$. Is $A$ still complete under the $\mathfrak{m}$-topology?
| https://mathoverflow.net/users/18119 | On the Completion of a complete local ring | I assume that $(R, \frak{m} )$ is a complete Noetherian local ring. Set $\frak{a} = \bigcap\_\lambda \frak{a}\_\lambda$. By passing to $R/\frak{a}$ we may assume that the $\frak{a}\_\lambda$-topology is separated. Now, we use a Theorem of Chevalley (1946) which says that in the complete Noetherian local ring the $\frak... | 5 | https://mathoverflow.net/users/17901 | 82362 | 49,288 |
https://mathoverflow.net/questions/82373 | 6 | Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
| https://mathoverflow.net/users/16858 | Mumford-Tate group and Galois representations | This is an immediate consequence of Theorem 2.11 and Proposition 2.9 in Deligne's 'Hodge cycles on abelian varieties' (notes by Milne available here: <http://www.jmilne.org/math/Books/DMOS.pdf>
2.11 shows that every Hodge cycle on an abelian variety $A$ over a field $k$ embeddable in $\mathbb{C}$ is absolutely Hodge,... | 12 | https://mathoverflow.net/users/7868 | 82386 | 49,296 |
https://mathoverflow.net/questions/82349 | 6 | Hi, I have a question about the positive Euler characteristic case. My question is: why is it so difficult as compared to the zero and negative cases? I am more interested in a pictorial/intuitive answer as compared to a very rigorous analytic answer. In other words, I want to get an intuitive "feel" of what goes wrong... | https://mathoverflow.net/users/19655 | uniformization theorem via ricci flows | I'm not an expert on Ricci flow but I believe the rough general reason for this is as follows. In dimension 2 the normalized Ricci flow gives the following evolution equation for scalar curvature
$$
\frac{\partial R}{\partial t}=\Delta\_t R+R(R-r)
$$
Where $r=\frac{\int\_MR}{vol M}=2\pi \chi(M)$. The analysis of this e... | 11 | https://mathoverflow.net/users/18050 | 82392 | 49,299 |
https://mathoverflow.net/questions/82390 | 5 | Let $R$ be a noetherian (commutative) ring. It is a well-known fact that for $R$ regular, $K$-theory of (finitely generated) projective modules and $K$-theory of arbitrary (f.g.) modules agree. Does the *converse* hold, i.e., suppose the natural map
$$K\_i(Proj R-Mod) \rightarrow K\_i(R-Mod)$$
is an isomorphism for... | https://mathoverflow.net/users/18116 | K-theory and regular rings | I don't know how to answer that question, but suppose we strengthen the hypothesis by making it apply also to all polynomial rings $R[t\_1,\dots,t\_n]$ over $R$, for $n \ge 0$. Then because $K\_i(R-Mod) \to K\_i(R[t]-mod)$ is an isomorphism (Quillen, Theorem 8, Higher Algebraic K-theory:I) for all noetherian rings $R$,... | 9 | https://mathoverflow.net/users/15247 | 82393 | 49,300 |
https://mathoverflow.net/questions/82376 | 11 | One can define twisted cohomology theories via bundles of classifying spaces. In particular, given a cohomology theory $h^{\*}$ and a corresponding $\Omega$-spectrum $E\_{n}, \varepsilon\_{n}$, we can consider on a space $X$ a bundle with fiber $E\_{n}$, and define a twisted version of $h^{n}$ as the set of homotopy cl... | https://mathoverflow.net/users/10758 | Twisted de-Rham cohomology and Eilenberg-Mac Lane spaces | You cannot compare both cohomologies unless $H$ has degree $1$, because otherwise the cohomology of $d+H\wedge$ is not $\mathbb{Z}$-graded.
If $H$ has degree $1$ then the cohomology of $d+H\wedge$ is the cohomology of $X$ with local coefficients corresponding to the flat line bundle with $1$-form $H$.
The only twis... | 8 | https://mathoverflow.net/users/12166 | 82394 | 49,301 |
https://mathoverflow.net/questions/82379 | 4 | A (finite dimensional) algebra is called biserial, if the radical of each projective indecomposable left/right module is the sum of two uniserial modules whose intersection is either trivial or simple.
It is known that a certain subclass of algebras (called gentle algebras) is closed under derived equivalence.
What... | https://mathoverflow.net/users/15887 | If an algebra is derived equivalent to a (special) biserial algebra is it biserial? | The answer to your question is no, in general. A simple counterexample is provided by a path algebra of the Dynkin quiver $D\_4$ and another algebra tilted from it. That is, let $A$ be the path algebra where the quiver $D\_4$ is oriented so that the vertex of degree 3 is a source and let $B$ be the quotient of the path... | 6 | https://mathoverflow.net/users/11791 | 82407 | 49,306 |
https://mathoverflow.net/questions/82408 | 9 | For the heat equation $(\partial\_t-\partial\_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the potential solution at infinity, i.e. $|f(t,x)|<\exp(c\cdot x^2)$.
My question is, if we do not impose any such condi... | https://mathoverflow.net/users/19673 | Non-uniqueness of solutions of the heat equation | Tychonoff in his 1935 paper [Théorèmes d'unicité pour l'équation de la chaleur](http://mi.mathnet.ru/eng/msb/v42/i2/p199) proved uniqueness if the solutions are not too large, and gave an example to show that the solution is not unique in general. His counterexample grows extremely rapidly for large x.
| 14 | https://mathoverflow.net/users/51 | 82410 | 49,308 |
https://mathoverflow.net/questions/82385 | 1 | I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression
$$
S = \sum\_i \frac{N\_i p\_i^i}{\sum\_j N\_j p\_i^j}
$$
in terms of the unknowns $N\_1, \dots, N\_t$, $p\_1, \dots, p... | https://mathoverflow.net/users/9896 | Bound on expression from probability distributions | $t$ is in fact a tight bound. It's slightly tricky because the objective is not defined at what should be the optimal solution (due to zeros in numerators and denominators).
What you want is first $p\_1 \to 0+$ (making the first term $ \to N\_1 p\_1/(N\_1 p\_1) = 1$,
then $N\_1 \to 0+$ making the second term $\to N\_2 ... | 1 | https://mathoverflow.net/users/13650 | 82414 | 49,311 |
https://mathoverflow.net/questions/82398 | 0 | I have the following problem: If $\Lambda$ is a hereditary, basic and connected algebra and $e$ is an idempotent of $\Lambda$, how can I prove that $e\Lambda e$ is also hereditary?
| https://mathoverflow.net/users/19671 | Hereditary algebras | If $\Lambda$ is split basic, then by Gabriel's theorem it is isomorphic to $\Bbbk Q$ where $Q$ is a finite acyclic quiver. Up to isomorphism you can assume $e$ is the sum of empty paths running over some subset $X$ of vertices. Then $e\Lambda e$ is isomorphic to the path algebra on the full (i.e. induced) subquiver on ... | 4 | https://mathoverflow.net/users/15934 | 82419 | 49,313 |
https://mathoverflow.net/questions/82416 | 8 | There are sets of points in $\mathbb{R}^n$ congruent to their own proper subsets. A (trivial) example is a ray, or to give a more interesting bounded example, $\{e^{i \cdot n} \mid n\in\mathbb{N}\}$. There are [references](http://goo.gl/JVzzy) saying about a proof by [Jan Mycielski](http://en.wikipedia.org/wiki/Jan_Myc... | https://mathoverflow.net/users/9550 | A set of points congruent to its proper subset | The finite case is pretty cheap. Take the free group $G$ with countably many generators $R\_1,R\_2,\dots$. Consider any injection $S:F(G)\to \mathbb N$ where $F(G)$ is the set of finite subsets of $G$ such that $S(W)$ is different from any index of a generator contained in a word $w\in W$ and take $E$ to be all (irredu... | 9 | https://mathoverflow.net/users/1131 | 82423 | 49,315 |
https://mathoverflow.net/questions/71883 | 8 | If I have a square matrix in $\mathbf{A} \in \mathbb{R}^{n \times n}$, I want to find another diagonal matrix $\mathbf{D} \in \mathbb{R}^{n \times n}$ that minimizes the residual $ \min\_\mathbf{D} || \mathbf{A-D} ||^2 $, where the norm here is the induced norm $\max\_{x\neq 0} \frac{\mathbf{||Ax||\_2}}{\mathbf{||x||\_... | https://mathoverflow.net/users/7268 | A Closed Form for the Diagonal Matrix Nearest an Arbitrary Square Matrix | The case of the $2$-norm may well not have a closed form. However, in the Frobenius norm the problem has a trivial answer: $D\_A = \mathrm{diag}(A)$. Since $\| A \|\_2 \leq \| A \|\_F \leq \sqrt{n} \| A \|\_2$, one then has that
$$
\frac{1}{\sqrt{n}} \| A - D\_A \| \leq \min\_{D} \| A - D \| \leq \| A - D\_A \|,
$$
whi... | 3 | https://mathoverflow.net/users/19011 | 82427 | 49,317 |
https://mathoverflow.net/questions/82370 | 20 | What is the distance in the sense of Gromov-Hausdorff between $\mathbb{Z}\_{p\_1}$ and $\mathbb{Z}\_{p\_2}$ with the usual p-adic metrics?
I got stuck and simply have no idea how to deal with such questions: I've got two metric trees and have to observe somehow all embeddings to all spaces which seems a bit intractable... | https://mathoverflow.net/users/13842 | Gromov-Hausdorff distance between p-adic integers. | The Gromov--Hausdorff distance is good only to define topology;
i.e., one should not care about distance between particular spaces.
But since you insist, I will answer an easier question which is closely related.
There is a modified distance $d'\_{GH}(X,Y)$ defined as infimum of all numbers $\varepsilon>0$ such that... | 12 | https://mathoverflow.net/users/1441 | 82428 | 49,318 |
https://mathoverflow.net/questions/82432 | 3 | Let $P$ be a smooth connected projective variety (say, over complex numbers); $H$ is its smooth hyperplane section. What can be said about the Zariski cohomology of $H$ with constant coefficients? It is certainly zero in positive degrees if $H$ is irreducible. Does it vanish in lower degrees (those that are smaller tha... | https://mathoverflow.net/users/2191 | Does the (torsion) Zariski cohomology of a (singular) hyperplane section of a smooth projective variety vanish (in small degrees)? | It does not necessarily vanish. For example, if $Y$ is the union of two irreducible curves meeting at more than one point and $A$ is a non-zero abelian group, $H^1(Y, A)$ is not zero.
| 2 | https://mathoverflow.net/users/4790 | 82433 | 49,319 |
https://mathoverflow.net/questions/82438 | 3 | Let $(M,hyp)$ be a closed hyperbolic surface. fix a point $m$ in $M$ and denote by $G=\pi\_1(M,m) $.
now let $\alpha$ and $\beta$ in $G$ such that $\alpha$ and $\beta$ does not commute . my first question is why $<\alpha,\beta>$ is of inifinite index in $G$ and my second question: Let $n$ be the least positive integer ... | https://mathoverflow.net/users/nan | non commutative elements in the fundamental group of a closed hyperbolic surface | The answer to the first question: because it is free, and every subgroup of finite index is the fundamental group of a finite cover of the surface (which is again a hyperbolic surface). The answer to the second question is "infinity". Take any free non-abelian subgroup $F$ of the surface group. The group $F$, being fre... | 4 | https://mathoverflow.net/users/nan | 82439 | 49,322 |
https://mathoverflow.net/questions/82453 | 3 | Let $X$ be a non-singular complex variety with a big line and base point free bundle $M$ on it. My question is can we say that for any locally free sheaf $F$ on $X$, $F \otimes M^n$ is globally generated for $n \gg 0$.
Motivation: If $M$ were an ample line bundle then all we need is that $F$ is coherent sheaf. But s... | https://mathoverflow.net/users/7455 | 'Ampleness' of a big line bundle | In general the answer is no, even $F$ is a line bundle itself. It is easy to see that a globally generated line bundle is nef, and if $F$ is not nef, and the segment between $F$ and $M$ does not intersect with the ample cone in $N^{1}(X)$, then $F \otimes M^{n}$ is numerically propotional to a divisor lies in the inter... | 6 | https://mathoverflow.net/users/18119 | 82454 | 49,327 |
https://mathoverflow.net/questions/82451 | 3 | It is well know that a 2 component link complement may doesn't detect the link type.
My question is **whether the following type of 2 component links detect their link types?**
**Such a link is composed of a knot with its meridian circle.** Here its meridian circle
means a meridian of the boundary of a solid torus n... | https://mathoverflow.net/users/19051 | a special type of 2 component link complement | I suppose that you are asking whether such links are determined by their complement. If this is the case, the answer is yes and it is a consequence of Gordon-Luecke's "knots are determinent by their complement" theorem.
The technique of a proof goes as follows. Since now "the complement of" means "the complement of ... | 6 | https://mathoverflow.net/users/6205 | 82460 | 49,331 |
https://mathoverflow.net/questions/82457 | 5 | Recall that the center $\mathrm{Z}(C)$ of a category $C$ is the monoid of endomorphisms of $\mathrm{id}\_C$. Thus $\eta \in \mathrm{Z}(C)$ is given by a familiy of endomorphisms $\eta\_x : x \to x$, where $x \in C$, such that for all morphisms $x \to y$ the obvious diagram commutes. The center of the category of rings ... | https://mathoverflow.net/users/2841 | Center of the category of special $\lambda$-rings | Your map is surjective too.
The free (special) $\lambda$-ring on one generator is a polynomial algebra of the form $F=\mathbb{Z}[\lambda^1(x),\lambda^2(x),\lambda^3(x),\dots]$. (This is well-known; I think Donald Yau proves it in his book on $\lambda$-rings.) The set of endomorphisms of the forgetful functor $\lambda... | 4 | https://mathoverflow.net/users/437 | 82477 | 49,342 |
https://mathoverflow.net/questions/82480 | 11 | An MSc student asked me if I knew an example of a prime $p$ and some finite layer $K\_n$ in the cyclotomic $\mathbf{Z}\_p$-extension of $\mathbf{Q}$ (so $[K\_n:\mathbf{Q}]=p^n$) which had non-trivial class group. My gut feeling was that fixing a $p$ and then going up the tower was a bad idea in the sense that going alo... | https://mathoverflow.net/users/1384 | Non-trivial class number at some finite level in the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$? | I found the notes of Coates' seminar I alluded to in my comment above. He said the following:
For $n \ge 1$, let $h(n)$ denote the class number of the unique cyclic degree $n$ extension contained in the compositum of all the cyclotomic $\mathbb{Z}\_p$-extensions for $p \mid n$. It is apparently not too difficult to ... | 12 | https://mathoverflow.net/users/2481 | 82488 | 49,347 |
https://mathoverflow.net/questions/82492 | 1 | Hi.
As is known, a polynomial $P \in K[x\_1, \dots, x\_n]$ is symmetric when permuting its variables always yields the same polynomial. This immediately yields an algorithm $O(n!)$ to check for symmetry of a polynomial.
Are there known algorithms faster than $O(n!)$ (perhaps using other bounds, like the degree) to... | https://mathoverflow.net/users/16913 | Nontrivial algorithm to check for polynomial symmetry? | One needs only check $n$ transpositions, since if each of the transpositions $(12),(23), \dots$ preserves a polynomial, then every permutation preserves that polynomial.
| 8 | https://mathoverflow.net/users/18060 | 82493 | 49,348 |
https://mathoverflow.net/questions/68946 | 10 | I am an analyst struggling through some geometry used in physics.
**Some background:** For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection 1-form on $P$ (a "principal connection"). This is a Lie algebra-valued 1-form.
As for the curvature two-form, e... | https://mathoverflow.net/users/4281 | Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P? | Your confusion is revealed in this sentence "Or one defines something called the exterior covariant derivative D (see wiki) and then the curvature is simply the exterior covariant derivative of the connection one-form." This is just not true; one does not take the `exterior covariant derivative of the connection $1$-fo... | 10 | https://mathoverflow.net/users/13972 | 82496 | 49,351 |
https://mathoverflow.net/questions/38863 | 9 | Let $\mathcal C$ and $\mathcal D$ be categories with suitable limits and colimits for the following discussion. Is it possible to re-interpret, or "re-seat" a monad $T : \mathcal C \to \mathcal C$ as a monad over $\mathcal D$? When $T$ is finitary, I know at least one way to do this. Compute the Kleisli category $\math... | https://mathoverflow.net/users/800 | Re-seating a monad | I like the question, Aleks - and I like the term "re-seating".
Yes, there is a direct way, at least in suitable circumstances. First suppose that we're beginning with a finitary monad $T$ on $\text{Set}$. For each set $X$, we have
$$
T(X) = \int^n T(n) \times X^n
$$
where the coend is over the category of finite sets... | 10 | https://mathoverflow.net/users/586 | 82497 | 49,352 |
https://mathoverflow.net/questions/82509 | 1 | I have a question about fixed points of Galois group actions.
I am hoping that this is easy for the experts.
Let $k$ be a field of characteristic $0$. Let $K$ be a finite
Galois extension of $k$ with Galois group $G$.
Supose that $A$ is any finite dimensional $G$-representation over
$k$. Then $G$ acts diagonally... | https://mathoverflow.net/users/15689 | Dimension of fixed points of Galois group actions | This follows from the fact that $K$ is isomorphic as $G$ module to the free module $k[G]$.
(use the existence of a basis of $K$ over $k$ consisting of Galois conjugates.
| 2 | https://mathoverflow.net/users/13992 | 82513 | 49,358 |
https://mathoverflow.net/questions/82481 | 9 | Consider $0=t\_0\leq t\_1\leq...\leq t\_n=1$, $f\_0,...,f\_{n-1}\in\mathbb{Z}$ and $F:[0,1]\to\mathbb{R}$ be such that
1) $F\equiv f\_i$ on the interval $(t\_i,t\_{t+1})$, for all $i=0,...,n-1$,
2) $\displaystyle \int\_0^1 F(t) dt=\sum\_{i=0}^{n-1}(t\_{i+1}-t\_i)f\_i=0$.
Does there exist an arbitrarily large pri... | https://mathoverflow.net/users/9736 | Piecewise constant functions with zero average | Thanks to the comments of George Lowther and Greg Martin (for which I am most grateful), I can now show that the answer is YES for infinitely many primes $q$.
**Theorem 1.** Let $t\_1\dots,t\_{n-1}$ be any finite set of real numbers.Then for any $\epsilon>0$ and any integer $r>0$ there are infinitely many primes $q\e... | 6 | https://mathoverflow.net/users/11919 | 82515 | 49,359 |
https://mathoverflow.net/questions/82510 | 7 | Let $\chi$ be a Dirichlet character, and define $\phi\_\chi (n)$ so that it satisfies $$\sum\_{n=1}^\infty \phi\_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
$$\phi\_{\chi}(n)=\sum\_{d|n}\mu\left(\frac{n}{d}\right)\chi\left(\frac{n}{d}\right)d=\left(\text{Id}\*\mu\chi\right)(n).$$
>
> My quest... | https://mathoverflow.net/users/12176 | The Correlation of the Möbius Function and Dirichlet Characters | Up to a multiplicative constant the answer is what you discovered, $f(n)=\sqrt{\log\log n}$. Obviously you can assume that $\chi$ is not a principal character since in that case you get something less than $1$. Then after writing
$$\sum\_{d|n}\frac{\mu (d)\chi(d)}{d}=\prod\_{p|n}\left(1-\frac{\chi (p)}{p}\right)$$and t... | 7 | https://mathoverflow.net/users/19368 | 82519 | 49,363 |
https://mathoverflow.net/questions/82476 | 12 | The classical computability theory taking place in $\mathbb{N}$, can be extended to more general spaces, like $T\_0$ second countable topological spaces $(X, \mathcal{O}, v)$ where $\mathcal{O}$ is a countable basis of $X$ and $v:\mathbb{N} \rightarrow \mathcal{O}$ a total surjection. Then we say that $f:X \rightarrow ... | https://mathoverflow.net/users/14490 | Effective topos and computability in topological spaces | There are many realizability toposes, of which the effective topos is just one. Each realizability topos has its own internal language in which topology and analysis can be developed to a considerable degree. Thus we get many notions of computable topology, not just one.
If you are interested in computable topology i... | 14 | https://mathoverflow.net/users/1176 | 82530 | 49,370 |
https://mathoverflow.net/questions/82527 | 9 | Given $2n$ points $x\_1, x\_2 \ldots x\_{2n}$ and a distance $d\_{i,j}$ defined between them, how can I best find the set $P$ of mutually exclusive pairs $(i,j)$ such that the sum of their distances
$$
\sum\_{(i,j) \in P} d\_{i,j}
$$
is minimised? The definition of $d\_{i,j}$ is open and the function could be conv... | https://mathoverflow.net/users/19699 | finding the $n$ closest pairs between $2n$ points | What you're looking for is a minimal weight perfect matching on a complete graph. From a quick wikipedia search, I think [Edmonds' matching algorithm](http://en.wikipedia.org/wiki/Edmonds%27s_matching_algorithm) will do the trick for you (see the bottom of the page) - I think it's polynomial time in the number of nodes... | 11 | https://mathoverflow.net/users/11054 | 82531 | 49,371 |
https://mathoverflow.net/questions/82533 | 0 | My question is whether for every extension of number fields $L\subset K$, and for every $f\_0(x),...,f\_n(x)$ in $K[x]$, there is some $\alpha\in L$ such that $$f\_n(\alpha)T^n+...+f\_1(\alpha)T+f\_0(\alpha)$$
is irreducible as a polynomial in $K[T]$.
If $L=K$ this is known from Hilbert's Irreducibility Theorem. I fi... | https://mathoverflow.net/users/5309 | A question related to Hilbert's Irreducibility Theorem | The answer is yes, assuming that the two-variable polynomial $f\_n(x)T^n + \dots + f\_1(x)T + f\_0(x)$ is irreducible over $K$.
This follows from the version of Hilbert's irreducibility theorem for number fields proved as Theorem 46 of p.298 of Schinzel's book *Polynomials with special regard to reducibility*: the re... | 3 | https://mathoverflow.net/users/16510 | 82535 | 49,373 |
https://mathoverflow.net/questions/81934 | 6 |
>
> Let $M$ be a natural number $M>1$. For every prime $p\_i$ not dividing $M$ take an arithmetic progression $A\_i=k\_i+np\_i$ , $n\geq 0$ such that $k\_i>p\_i^2/M$. Is there any $M$ and some choice of the $k\_i's$ such that $\cup A\_i= \mathbb{N}$ ?
>
>
> CONJECTURE: There is not such choice for any $M$
>
>
>
... | https://mathoverflow.net/users/14726 | Generalising Dirichlet's theorem in arithmetic progressions-prime combinatorics | I can't resist some comments. I find it highly unlikely that what you ask is possible. The question seems a bit odd in that it is so specific when more basic questions are open.
In the following some estimates are rough and corrections are welcome. Let $p\_i$ be the $i$th prime and $P\_t$ the product of the first $t... | 4 | https://mathoverflow.net/users/8008 | 82537 | 49,374 |
https://mathoverflow.net/questions/82538 | 1 | What is the shortest formal statement you can write that is provably equivalent to the Continuum Hypothesis in ZFC?
Please use only variables and the following symbols: $\forall, \exists,\lor,\land,\neg,\to, \in,=$ (parentheses may be added for convenience and do not contribute to the length of the formula). For exa... | https://mathoverflow.net/users/9550 | Shortest formal statement equivalent to the continuum hypothesis | I don't know if this is the shortest (number of symbols?) but in
D. Scott, "A Proof of the Independence of the Continuum Hypothesis", THEORY OF COMPUTING SYSTEMS, Volume 1, Number 2. Available at: [http://www.springerlink.com/content/hh339022jt1m5183/](https://doi.org/10.1007/BF01705520)
there is (at the bottom of ... | 6 | https://mathoverflow.net/users/11618 | 82544 | 49,377 |
https://mathoverflow.net/questions/82512 | 3 | Let $X$ be a scheme which is smooth over a noetherian base scheme $S$.
Let $(\mathcal F,\nabla)$ be a flat vector bundle, i.e. $\mathcal F$ is a vector bundle of finite rank on $X$ and $\nabla$ is an integrable connection relative $S$.
Is there a notion in the literature of "unipotent" flat vector bundle and morphism... | https://mathoverflow.net/users/18183 | Unipotent vector bundles | As [Keerthi Madapusi Pera](https://mathoverflow.net/users/7868/keerthi-madapusi) points out in his comments, it is certainly reasonable to define a unipotent flat vector bundle as a flat vector bundle that is a successive extension of the trivial one $(\mathcal O\_X,d)$. Over a general basis $S$ I don't know, but over ... | 8 | https://mathoverflow.net/users/11682 | 82550 | 49,382 |
https://mathoverflow.net/questions/82320 | 8 | Recall the following theorem of Linton:
A functor $U:E\to \operatorname{Set}$ is monadic if
1. $U$ has a left adjoint,
2. $E$ admits kernel pairs and coequalizers,
3. A parallel pair $R \rightrightarrows S$ in $E$ is a kernel pair if and only if its image under $U$ is so, and
4. A morphism $A\to B$ in $E$ is a reg... | https://mathoverflow.net/users/1353 | The (co)monadicity theorem relative to a presheaf topos | The result you are after is Theorem 1.2 of "A Monadicity Theorem" by Borceux and Day, published in the Bulletin of the Australian Mathematical Society in 1977, and the paper is available online. If you have access to Mathscinet reviews but not the paper you can even see the result stated in detail in the review by Kell... | 7 | https://mathoverflow.net/users/17696 | 82553 | 49,384 |
https://mathoverflow.net/questions/82547 | 6 | Let $G$ be a finite group of order $n$ and $\psi(G)$ be the sum of element orders of $G$. Then $\psi(G)\leq\psi(C\_n)$, where $C\_n$ is the cyclic group of order $n$ (see "Sums of element orders in finite groups", Comm. Algebra 37 (2009), 2978-2980). Is it true a similar inequality for the product of element orders of ... | https://mathoverflow.net/users/17565 | A question on the product of element orders of a finite group | Denoting the order of $g$ by $o(g)$, you can show that for any decreasing function $f$ the following inequality holds
$$\sum\_{g\in G}f(o(g))\geq \sum\_{g\in \mathbb Z/n\mathbb Z}f(o(g)).$$
This is because one can actually construct a bijection $\sigma:G\to\mathbb Z/n\mathbb Z$ which satisfies $$o(\sigma(g))\geq o(g)$$... | 25 | https://mathoverflow.net/users/2384 | 82554 | 49,385 |
https://mathoverflow.net/questions/82445 | 4 | 1) Is there a criterion for telling if a Lie-algebra-valued $2$-form (for example on a $SU(2)$-bundle) is a curvature, without taking derivatives? For example, using Bianchi's identity is not allowed.
2) Also partial criteria are welcome (i.e. ones in which just a necessary/sufficient condition, formulated without re... | https://mathoverflow.net/users/5628 | criterion for being a curvature | This doesn't answer the original question; instead, it argues that the prohibition against using *any* derivatives of the 'candidate' curvature form is not really necessary. The first step is to show that, if a continuous $\frak{g}$-valued $2$-form $\Phi$ is the curvature of a continuous $\frak{g}$-valued $1$-form $\al... | 6 | https://mathoverflow.net/users/13972 | 82571 | 49,394 |
https://mathoverflow.net/questions/82500 | 8 | Hi,
is there a free group action of the cyclic group $\mathbb{Z}/n\mathbb{Z}$ on the infinite dimensional projective space $\mathbb{CP}^\infty$ for every $n\in \mathbb{N}$? And if there is one, how does it work?
Thanks
| https://mathoverflow.net/users/19695 | Free group actions on complex projective spaces | I believe I can give a complete answer.
First, let me collect several earlier comments by myself, Dylan Wilson and Alain Valette. When $N<\infty$ then there can not be a free action of $\mathbb Z\_n$ on $\mathbb{CP}^N$ when $n>2$. Indeed, if there were such an action then the square of the generator would act trivially... | 10 | https://mathoverflow.net/users/18050 | 82583 | 49,399 |
https://mathoverflow.net/questions/82541 | 2 | Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore finite. From an earlier post on MO ([Riemann surfaces that are not of finite type](https://mathoverflow.net/questions/78984... | https://mathoverflow.net/users/36038 | Hurwitz's automorphisms theorem for infinite genus Riemann surfaces | The action is discrete if $X$ is hyperbolic and is not a disk or annulus. By uniformization,
$X=\mathbb{H}^2/\Gamma$ for some discrete subgroup $\Gamma< PSL\_2(\mathbb{R})$.
Let $\Lambda < PSL\_2(\mathbb{R})$ be the normalizer of $\Gamma$, then
$Aut(X)\cong \Lambda/\Gamma$ since any conformal automorphism of $X$ must
... | 10 | https://mathoverflow.net/users/1345 | 82584 | 49,400 |
https://mathoverflow.net/questions/82399 | 8 | Notation:
* $k, m, n$ are non-negative integers
* $f, g, h$ are functions $\mathbb{N} \to \mathbb{N}$
* $f^k$ is $k$-th iterate of the function $f$: $f^0(n)=n, f^{k+1}(n)=f^k(f(n))$
* $f \prec g$ means eventual domination: $\exists\_m \forall\_{n>m} f(n) < g(n)$.
Let $S$ be the minimal set of functions $\mathbb{N} ... | https://mathoverflow.net/users/9550 | How this set of functions is ordered? | Here is a trivial way in which it is not well-ordered: the ordering used is not total (we have a preorder, rather than a total order). This is because we can construct a function $Z(n)$ given by $Z(0)=1$ and $Z(n)=0$ for $n>0$. Simply take, in the combination law, $f(n)=0$, $g(n)=n$, and $h(n)=1$. Then $Z$ and $0$ are ... | 7 | https://mathoverflow.net/users/5583 | 82587 | 49,401 |
https://mathoverflow.net/questions/82581 | 10 | I could not answer or find references of this question, even for the following special case:
On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function with integral $1$ the Jacobian of some diffeomorphism?
An equivalent formulation of the question is: On $S^2$, is eve... | https://mathoverflow.net/users/17294 | On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism? | Here is an "answer-version" of my comment:
Yes, this is true in general. The reference I know is Moser's 1965 paper "On the volume elements on a manifold" (<http://www.jstor.org/stable/1994022>).
Specifically, let $M$ be a compact connected orientable manifold, and let $\sigma$ and $\tau$ be smooth volume forms on ... | 17 | https://mathoverflow.net/users/2819 | 82590 | 49,403 |
https://mathoverflow.net/questions/82505 | 9 | Let M be a closed spin manifold of dimension $d$. One form of the elliptic genus of $M$ is
$$ F(q)=q^{-d/8} \hat A(M) {\rm ch} \otimes\_{k=1/2,3/2,\cdots} \Lambda\_{q^k}T \otimes\_{\ell=1}^\infty S\_{q^\ell}T [M] $$
where the notation follows that of E. Witten, ``The Index of the Dirac Operator in Loop Space."
The coef... | https://mathoverflow.net/users/10475 | Elliptic genus for manifolds with boundary | This, or a similar variant has been studied in [Secondary Invariants for String Bordism and tmf](http://arxiv.org/abs/0912.4875) and [The f-invariant and index theory](http://arxiv.org/abs/0808.0257). The deviation from modularity caused by the boundary gives the interesting invariant of the boundary.
| 10 | https://mathoverflow.net/users/7530 | 82600 | 49,407 |
https://mathoverflow.net/questions/82597 | 10 | **What is the geometric meaning of $\omega=dx/(2y+a\_1x+a\_3)$ for an elliptic curve?**
This question is an adjunct to MO [Q1](https://mathoverflow.net/questions/52241/formal-group-laws-and-l-series) on formal laws and L-series. Silverman (Q1) and [Darmon](http://www.math.mcgill.ca/darmon/pub/Articles/Research/36.NSF... | https://mathoverflow.net/users/12178 | Geometric picture of invariant differential of an elliptic curve | A paper by [John Tate](http://www.kryakin.com/files/Invent_mat_(2_8)/23/23_01.pdf) (pg. 1 and 2) gives a clear derivation of the diff. form:
Reparametrize the elliptic curve
$y^2+a\_1xy+a\_3y=x^3+a\_2x^2+a\_4 x+a\_6$
with $p(z)=x+(a\_1^2+4a\_2)/12$ and $p^{'}(z)=2y+a\_1x+a\_3$ to obtain
$(p^{'})^2=4p^3-g\_2p-g\... | 9 | https://mathoverflow.net/users/12178 | 82612 | 49,415 |
https://mathoverflow.net/questions/82603 | 4 | Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:
1) With orders $d\_1, d\_2, \ldots, d\_k$ in such a way that $d\_i|d\_{i+1}$,
2) With orders that are powers of not necessarily distinct primes $p\_1^{\alpha\_1}, \ldots, p\_n^{\alpha\_n}$.
Is it true, and how ca... | https://mathoverflow.net/users/19452 | Minimal generation for finite abelian groups | Note that $n$ is the sum over prime divisors $p$ of $|G|$ of the minimal number of generators of the distinct Sylow $p$-subgroups of $G.$ The sizes of all minimal generating sets of a finite $p$-group are the same by properties of the Frattini subgroup. Use of the Frattini subgroup helps to prove the leftmost inequalit... | 6 | https://mathoverflow.net/users/14450 | 82618 | 49,418 |
https://mathoverflow.net/questions/82485 | 10 | Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is there a notion of an ideal class group of a Dedekind ring spectrum (Picard group)?
Thanks
PS This question was alread... | https://mathoverflow.net/users/11546 | Dedekind spectra | In trying to generalize concepts from algebra to spectra, there are several issues that come into play.
In order for a concept in stable homotopy theory to be intrinsically meaningful it generally needs to be invariant under weak equivalence - whatever the appropriate notion of "weak equivalence" is (of spectra, of c... | 17 | https://mathoverflow.net/users/360 | 82624 | 49,420 |
https://mathoverflow.net/questions/82617 | 1 | hello community.
I need some help
given $k = \sum A\_q e^{ia\_q s}$ where $k$, and $s$ is known. Can $A\_q$ s now be expressed in terms of $a\_q$ .
any help in that direction will be appreciated.
| https://mathoverflow.net/users/19724 | exponential sum - general cases | No, you cannot, at least not at this level of generality. Consider for example $a\_q = \frac{\pi}{q}$, $s=1$ and $k=0$. Basically a roots-of-unity situation; then each $A\_q$ can be a multiple of the proper coefficient of a cyclotomic polynomial. That does 'solve' the problem, in that you get the $A\_q$'s in terms of t... | 1 | https://mathoverflow.net/users/3993 | 82626 | 49,421 |
https://mathoverflow.net/questions/82634 | -1 | A differentiable transformation of R^n at each point has an invertible derivative. Does it imply that the transformation is a global diffeomorphism?
| https://mathoverflow.net/users/16425 | Global invertibility | Let $f:\mathbb{R}^n\rightarrow M\subseteq\mathbb{R}^n$ be the differentiable transformation, with $M=f(\mathbb{R}^n)$. If $M\neq\mathbb{R}^n$, then obviously $f$ isn't an global diffeomorphism of $\mathbb{R}^n$. But it is global diffeomorphism between $\mathbb{R}^n$ and $M$.
---
\**This was intended to be a comme... | 0 | https://mathoverflow.net/users/17682 | 82641 | 49,426 |
https://mathoverflow.net/questions/82642 | 3 | Let $(P, \leq)$ be a total ordering (some of you prefer the name *linear order*). Can we find a subset $R\subseteq P$ which is well ordered (with respect to $\leq\upharpoonright R$) and cofinal in $P$, that is for each $p\in P$ there is $r\in R$ such that $p\leq r$?
| https://mathoverflow.net/users/19732 | Well-ordered cofinal subsets | Yes, it is called the [*cofinality*](http://en.wikipedia.org/wiki/Cofinality) of the order. Just apply Zorn's lemma to the class of all well-ordered suborders of $P$, ordered by end-extension. If you've got a maximal such order, then it must be cofinal, since otherwise you could end-extend it. Note that Zorn's lemma ap... | 7 | https://mathoverflow.net/users/1946 | 82645 | 49,429 |
https://mathoverflow.net/questions/82625 | 4 | The *Leray number* $L\_{\Bbbk}(K)$ (relative to a field $\Bbbk$) of a simplicial complex $K$ is the least $d\geq 0$ such that $\widetilde H\_n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$ of $K$.
Leray numbers have historically arisen in at least two distinct contexts. In combinatorics they arose i... | https://mathoverflow.net/users/15934 | Reference request on Leray numbers | For question 1, the earliest reference I know is:
Ralf Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988), Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 57–70.
(This actually proves an equivalent result on linear resolutions.) It seems to be the kind of thing that gets rediscovered s... | 5 | https://mathoverflow.net/users/19729 | 82650 | 49,433 |
https://mathoverflow.net/questions/82648 | 14 | [Apparently B6 of the Putnam](https://math.stackexchange.com/questions/88188/asking-2011-putnam-b6) this year asked:
>
> Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ of the numbers $\sum^{p-1}\_{k=0} k! n^{k}$ are not divisble by $p$.
>
>
>
With some rearrangeme... | https://mathoverflow.net/users/12176 | Truncated Exponential Series Modulo $p$: Deeper meaning for a Putnam Question. | I've seen this trick in a paper of Mit'kin, Math Zametki 1992. There he improves the bound. This is related to Stepanov's method to bound the number of solutions of equations over finite fields.
| 15 | https://mathoverflow.net/users/2290 | 82652 | 49,434 |
https://mathoverflow.net/questions/82578 | 4 | In a locally cartesian closed category $\mathcal C$, for every map $f:A\to B$, there is an associated pullback functor $f^\* : \mathcal C/B \to\mathcal C/A$. Moreover, if $g:B\to C$, the two functors $(g\circ f)^\*$ and $f^\*\circ g^\*$ are *canonically isomorphic*, but they have no reason to be "equal". Even in the ca... | https://mathoverflow.net/users/10217 | Functorial choice of pullbacks in a locally cartesian closed $(\infty,1)$-category | The easiest thing to do is probably to appeal to the equivalence between functors $\mathcal{C}^{\text{op}} \to \text{Cat}\_{\infty}$ and Cartesian fibrations over $\mathcal{C}$. Roughly speaking, a Cartesian fibration $\mathcal{E} \to \mathcal{C}$ corresponds to a functor that sends $c \in \mathcal{C}$ to the fibre $\m... | 4 | https://mathoverflow.net/users/1100 | 82658 | 49,438 |
https://mathoverflow.net/questions/82661 | 26 | Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well structured, well motivated, and perhaps with application to other fields.
any such book exists?
I tried a book by nad... | https://mathoverflow.net/users/14105 | Book recommendation for ergodic theory and/or topological dynamics? | I think another good choice is the book "Ergodic Theory: With a View Towards Number Theory" by Manfred Einsiedler and Thomas Ward,Graduate Texts in Mathematics 259.Besides basic concepts of ergodic theory,the book also discusses the connection between ergodic theory and number theory,which is a hot topic recently.And a... | 19 | https://mathoverflow.net/users/19702 | 82670 | 49,444 |
https://mathoverflow.net/questions/81649 | 1 | Let $C$ be a connected filtered coalgebra over a field $k$. Maybe $k$ has characteristic $0$ (though I don't know where this can be of use). Let $1$ denote the unique element of $C\_0$ mapping to $1\in k$ under the counit. Let $\overline{C^{\ast}}$ be the ideal of the dual algebra $C^{\ast}$ which consists of all $f\in... | https://mathoverflow.net/users/2530 | Linear functional kills all primitives of a connected filtered coalgebra => it lies in m^2? | Okay, I should have thought a bit more about this question before posting it on MathOverflow: the answer is No, for a combination of rather straightforward reasons. Moreover, the answer is No even for a locally finite connected graded coalgebra over a field $k$, where "locally finite" means that every graded component ... | 2 | https://mathoverflow.net/users/2530 | 82678 | 49,449 |
https://mathoverflow.net/questions/82664 | 10 | If we consider a partition ${\mathcal X}=(X\_1,\dots,X\_n)$ of a finite set $X$ and a partition ${\mathcal Y}=(Y\_1,\dots,Y\_m)$ of another finite set $Y$, then $(X\_i\times Y\_j)\_{i=1,...,n,j=1,...,m}$ is a partition of $X\times Y$. I will call this partition the "product partition" ${\mathcal X}\times{\mathcal Y}$.
... | https://mathoverflow.net/users/18938 | Partitions and finite sets | Here are comments but not a solution. Boris points out that the problem reduces to this: given a list of positive integers $c\_1 \dots c\_n$ are there lists $a\_1 \cdots a\_u$ and $b\_1\cdots b\_v$ so that the products $a\_ib\_j$ are the $m$ values. One problem is finding the values $u,v$ with $uv=n$ let us assume that... | 3 | https://mathoverflow.net/users/8008 | 82679 | 49,450 |
https://mathoverflow.net/questions/82676 | 14 | For an inverse system {$G\_i$} of finite groups, and a fixed field $\mathbb{k}$, one can consider the corresponding group algebras $\mathbb{k}[G\_i]$. The latter form an inverse system of $\mathbb{k}$-algebras {$\mathbb{k}[G\_i]$} (unless I miss something obvious). Is it true that the inverse limit of {$\mathbb{k}[G\_i... | https://mathoverflow.net/users/11100 | inverse limits of group algebras and profinite groups | See section 5.3 in "Profinite Groups" by Ribes and Zalesskii. The inverse limit of group algebras you are referring to is called the *complete group algebra* and it is the completion of the ordinary group algebra $\mathbb k[G]$ with its natural profinite topology. In other words $\mathbb k[G]$ is densely embedded in th... | 11 | https://mathoverflow.net/users/2384 | 82680 | 49,451 |
https://mathoverflow.net/questions/82704 | 3 | According to ESCARDÓ-LAWSON-SIMPSON paper 'Comparing cartesian closed categories of (core) compactly generated spaces' The following four propositions are true:
* A topological space $X$ is exponentiable iff $X$ is core-compact.
* The category of core-compact topological spaces is not cartesian closed, because even t... | https://mathoverflow.net/users/14490 | Why the category of core-compact spaces with continuous maps is not cartesian closed ? | Andrej and Qiaochu are right. Let me elaborate. If X and Y are core-compactly generated (ccg for short), then the exponential in the category of topological spaces need not exist (it exists iff X is core compact). But the exponential always exists in the category CCG. You have to take into account the facts that (1) th... | 12 | https://mathoverflow.net/users/19751 | 82706 | 49,467 |
https://mathoverflow.net/questions/82685 | 6 | Let *C* be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over *C* which presents the $(\infty,1)$-presheaf category of *C*? Ideally, such a model structure would be Quillen equivalent to the c... | https://mathoverflow.net/users/49 | Presheaves on a complete Segal space | Yes. Let $W$ be a complete Segal space, thought of as a simplicial "space" $(W\_q)$. The fibrant objects of your model category will be the fibrations $f:X\to W$ such that for each simplicial operator $\delta:[q]\to [p]$ with $\delta(q)=p$, the evident map from $X\_p$ to the pullback of
$$X\_q \xrightarrow{f} W\_q \xl... | 5 | https://mathoverflow.net/users/437 | 82711 | 49,470 |
https://mathoverflow.net/questions/82708 | 10 | In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C\*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary of concepts between these two objects. More specifically, given a compact Hausdorff space $X$, I ask in what manner are... | https://mathoverflow.net/users/19313 | Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X? | Gelfand duality asserts that $C(-)$ is an anti-equivalence from the category of compact hausdorff spaces to the category of commutative unital $C^{\ast}$-algebras. Now, for a continuous map $f : X \to Y$ it is not hard to show that
$f$ is surjective $\Longleftrightarrow$ $C(f) : C(Y) \to C(X)$ is injective
Sketch o... | 20 | https://mathoverflow.net/users/2841 | 82713 | 49,472 |
https://mathoverflow.net/questions/82701 | 4 | Let $\Omega\subset \mathbb{R}^n$ an open contractible set (we can assume $n=2$ for a start) and $\omega$ be a 1-form on $\Omega$ which is nowhere zero. Then $\omega=df$ for a function $f$ if and only if $d\omega=0$. If $\omega$ is not closed it might still be possible that it works up to a positive multiple. In other w... | https://mathoverflow.net/users/8794 | Existence of integrals of 1-forms up to multiples | Well, the local condition that $\omega\not=0$ be a nonzero multiple of an closed $1$-form is that $\omega\wedge d\omega = 0$. This is necessary and sufficient for the local existence of functions $f$ and $g\not=0$ such that $\omega = g\ df$. (This claim is just a special case of the Frobenius Theorem.)
For the global... | 10 | https://mathoverflow.net/users/13972 | 82715 | 49,473 |
https://mathoverflow.net/questions/82719 | 2 | Well, I know that I'm going to have some vote-down, because this should be a very simple property but the real story is that... I am not able to prove it!
For any fixed $n\in\mathbb N$, I have a finite partition of the natural numbers
$$
\mathbb N=A\_1^n\cup...\cup A\_{k(n)}^n
$$
(every set of the partition is in... | https://mathoverflow.net/users/13809 | Simple set-theoretic property to extract a subsequence in infinitely many steps | As requested, here's the comment posted as an answer:
What if the $n$th partition splits the naturals into ten pieces based on the $n$th digit of the decimal expansion?
(Of course ten is chosen for familiarity and not for optimality.)
---
Apologies for such an extensive edit, but I couldn't visualize this eit... | 12 | https://mathoverflow.net/users/14913 | 82723 | 49,476 |
https://mathoverflow.net/questions/82720 | 17 | The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm.
If we apply this to $\ell^2(R)$, then we see that $C^0([0;1],R)$ has a subspace which is a Hilbert space for the sup n... | https://mathoverflow.net/users/5743 | Banach-Mazur applied to a Hilbert space | Let $\varphi$ be a continuous function from $[0,1]$ onto the closed unit ball $B$ of $\ell^2$ in the weak topology. Then we define $J: \ell^2 \to C[0,1]$ by $(Jx)(t) = <\phi(t), x>$. Now, how to construct $\varphi$?
Consider convex compact subsets of $B$ of the form $K(a\_1,\ldots,a\_n) = \{x \in B: a\_i/2^n \le x\_... | 6 | https://mathoverflow.net/users/13650 | 82737 | 49,486 |
https://mathoverflow.net/questions/82730 | 2 | "Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The original proof used modular functions.
I'm wondering if this has been generalized to other open complex manifolds. I do... | https://mathoverflow.net/users/11286 | Little Picard for (open) complex manifolds? | This paper seems to address this question in considerable generality:
Some Picard Theorems for Holomorphic Maps to Algebraic Varieties
by ML Green - 1975 -
(it is also very widely cited, so presumably is not the last word on the subject).
| 1 | https://mathoverflow.net/users/11142 | 82738 | 49,487 |
https://mathoverflow.net/questions/82750 | 4 | This is likely a very easy counting question inspired by some elementary geometry:
Consider a simple rectilinear polygon embedded in a plane in such a way that each of its edges is parallel to one of the coordinate axis. Two such polygons are considered *distinct* if they are not related by some composition of transl... | https://mathoverflow.net/users/16504 | Counting rectilinear polygons | The number of right turns must always be 4 more than the number of left turns. This is the only constraint on turn sequences. Proof: Take the minimal impossible sequence satisfying this property. It must contain RR, so it is either RRRR or contains RRL. Delete the RL, find an example, and then add a small "tab" off the... | 3 | https://mathoverflow.net/users/18060 | 82763 | 49,501 |
https://mathoverflow.net/questions/82739 | 2 | Hello
I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor polynomials for any point of expansion $x\_i \in (a,b)$ of finite order $N\_i$,
$$
\sum\_{n=0}^{N\_{i}}\frac{f^{(n)}(x\_{i})}... | https://mathoverflow.net/users/9404 | Approximation:- Algorithmic considerations | First of all, if you need a polynomial approximation, I'd rather rescale $f$ to put it on $(-1,1)$ and interpolate it at the Chebyshev nodes <http://en.wikipedia.org/wiki/Chebyshev_nodes>. Furthermore, instead of using the monomial basis, as for Taylor polynomials, it is better to expand the function in a series of Che... | 3 | https://mathoverflow.net/users/7482 | 82767 | 49,505 |
https://mathoverflow.net/questions/82067 | 2 | I need an example of a CR submanifold of maximal CR dimension with the shape operator of the distinguished normal equals zero, or a hypersufrace of the shape operator equals zero. Can anyone help me?
| https://mathoverflow.net/users/19582 | shape operator of a hypersurface or a CR submanifolds of maximal CR dimension | Well, there are examples, but it appears that they are all essentially trivial.
Take the simplest case of a curve in $\mathbb{C}^n$, which automatically has maximal CR dimension. (Of course, that CR dimension is $0$). Let's say that it is parametrized at constant speed, $\gamma:(a,b)\to \mathbb{C}^n$. Then the condit... | 3 | https://mathoverflow.net/users/13972 | 82794 | 49,516 |
https://mathoverflow.net/questions/82790 | 2 | Let $S$ be a $K3$ surface. Is it true that any sheaf on $S$ with zero Chern classes is isomorphic to $\mathcal{O}\_S^{\oplus n}$ for some $n$? If not, do you have any counterexample?
| https://mathoverflow.net/users/33841 | Sheaves with zero Chern classes on a $K3$ surface. | The answer is **no**. Here is a counterexample.
Take an ample divisor $L$ on $S$ and let $Z \subset S$ be a zero-dimensional subscheme of length $\ell(Z)=L^2$.
Now consider the coherent sheaf $$\mathscr{F}=\mathscr{O}\_S(-L) \oplus \mathscr{O}\_S(L) \otimes \mathscr{I}\_Z.$$
Straightforward computations show that... | 6 | https://mathoverflow.net/users/7460 | 82796 | 49,517 |
https://mathoverflow.net/questions/82781 | 0 | Can anyone explain to me what the Hodge decomposition form of a symplectic form in a special symplectic manifold looks like?
| https://mathoverflow.net/users/19582 | Hodge decomposition of a symplectic form. | Using the additional information that the OP provided in the comments to Yael Fregier's answer, I can elaborate as follows:
I still don't know what "special complex manifold" means, but in any case, I will assume the following. If $(M, J, \nabla)$ is a complex manifold with a connection $\nabla$ coming from a metric ... | 1 | https://mathoverflow.net/users/6871 | 82797 | 49,518 |
https://mathoverflow.net/questions/82792 | 8 | Hello,
Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?
Thank you,
Sasha
| https://mathoverflow.net/users/2095 | Proper morphism sending coherent to coherent | Gerd Faltings, Finiteness of coherent cohomology for proper fppf stacks, J. Algebraic Geometry 12 (2003) 357–366
| 4 | https://mathoverflow.net/users/2035 | 82803 | 49,520 |
https://mathoverflow.net/questions/82770 | 21 | I know of two places where $K\_{\*}(\mathbb{Z}\pi\_{1}(X))$ (the algebraic $K$-theory of the group ring of the fundamental group) makes an appearance in algebraic topology.
The first is the Wall finiteness obstruction. We say that a space $X$ is finitely dominated if $id\_{X}$ is homotopic to a map $X \rightarrow X$... | https://mathoverflow.net/users/9481 | Algebraic K-theory of the group ring of the fundamental group | To add to Tim Porter's excellent answer:
The story of what we now call $K\_1$ of rings begins with Whitehead's work on simple homotopy equivalence, which uses what we now call the Whitehead group, a quotient of $K\_1$ of the group ring of the fundamental group of a space.
On the other hand, the story of $K\_0$ of ... | 18 | https://mathoverflow.net/users/6666 | 82807 | 49,523 |
https://mathoverflow.net/questions/82804 | 4 | Let $k$ be a field with $p$ elements. Consider the following computational problem
>
> *Input*: a natural number $n$, $n^2$ linear forms $M\_{ij}$, $i,j=1,\ldots n$ in $n^2$ variables $X\_{11}, \ldots X\_{nn}$.
>
>
> *Problem:* Is there an assignement of values to the variables $X\_{ij}$ so that the matrix $M\_{... | https://mathoverflow.net/users/2631 | determining if a matrix of linear forms represents a non-degenerate matrix | The determinant of $M$, considered as a matrix over the polynomial ring $R=k[X\_{11},\dots,X\_{nn}]$, is a polynomial $f\in R$, and your problem is to determine whether $f$ defines the constant $0$ function over $k$.
There are several division-free algorithms for computation of determinant in any commutative ring usi... | 5 | https://mathoverflow.net/users/12705 | 82808 | 49,524 |
https://mathoverflow.net/questions/82819 | 8 | Can you give me an example of a finitely generated infinitely presented amenable group which is a quotient of a finitely presented amenable group?
| https://mathoverflow.net/users/7307 | Quotients of f.p. amenable groups | Take the finitely presented solvable group $G$ with undecidable word problem, constructed by Kharlampovich. That group has infinite center that is a direct product of infinite number of cyclic group. The center has uncountably many subgroups $N\_\alpha$, each normal in $G$, most groups $G/N\_\alpha$ are not finitely pr... | 12 | https://mathoverflow.net/users/nan | 82820 | 49,530 |
https://mathoverflow.net/questions/82817 | 0 | Let $X\_i$ be a sequence of i.i.d. $\mathbb{R}^d$-valued, continuous (i.e. with density) random variables. We assume that $E X\_i =0$ and $Cov(X\_i)=Id$. Let
$S\_n:=\frac{1}{\sqrt{n}}\sum\_{i=1}^n X\_i.$
For $d=1$, under assumption of existence of the $k$-th moment, it is known that
$p\_n(x)=g(x) + f\_k(x) +o(n... | https://mathoverflow.net/users/1302 | Multivariate CLT, convergence of densities | The answer to your first question is yes. See Chapter 19 of [\*Normal Approximation and Asymptotic Expansions](http://books.google.com/books?id=H1lOIVHcRDEC&lpg=PP1&dq=bhattacharya%2520and%2520rao&pg=PP1#v=onepage&q&f=false) by Bhattacharya and Rao.
I believe the answer to your second question is no, but don't know w... | 2 | https://mathoverflow.net/users/1044 | 82822 | 49,531 |
https://mathoverflow.net/questions/82813 | 12 | Call a diagram $E$ in a model category a *homotopy colimit diagram* if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A *homotopy colimit* is defined as the categorical colimit of a cofibrant replacement of the diagram in the projective model structure and this is where the morphism com... | https://mathoverflow.net/users/19548 | Does the right adjoint of a Quillen equivalence preserve homotopy colimits? | The homotopy colimit functor $Ho(D^I)\rightarrow Ho(D)$ is the left adjoint of the constant diagram functor $Ho(D)\rightarrow Ho(D^I)$. Quillen equivalences induce Quillen equivalences between diagram categories, you you can replace $D$ with $C$, hence you're done by uniqueness of adjoints.
PS Don't worry about the f... | 6 | https://mathoverflow.net/users/12166 | 82824 | 49,532 |
https://mathoverflow.net/questions/82825 | 7 | Let $\mathcal{K}(\mathcal{H})$ be the C\*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c\_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many copies of this algebra.
Is it \*-isomorphic to $\mathcal{K}(\mathcal{H})$ itself? Or at least as a Banach space?
| https://mathoverflow.net/users/19732 | $c_0$-direct sum of $\mathcal{K}(\mathcal{H})$ | Yes, there is a Banach space isomorphism. The $c\_0$ sum of $\mathcal{K}(H)$ is clearly isometrically isomorphic to its own $c\_0$ sum and contains $\mathcal{K}(H)$ as a norm one complemented subspace, so by the Pelczynski decomposition method it is enough to observe that the $c\_0$ sum of $\mathcal{K}(H)$ embeds into ... | 11 | https://mathoverflow.net/users/2554 | 82831 | 49,534 |
https://mathoverflow.net/questions/82830 | 6 | Given an affine algebraic variety $V$ such that $\Gamma(V,\mathcal{O}\_V)$ is a UFD, its sheaf of ring can be determined easily since one can show that:
$$\Gamma(D(f\_1) \cup \cdots \cup D(f\_n),\mathcal{O}\_V) \simeq \Gamma(D(h),\mathcal{O}\_V),$$
where $h=\mathrm{gcd}(f\_1, \ldots, f\_n)$.
It is then natural to a... | https://mathoverflow.net/users/14037 | Algebraic varieties and UFD | If and only if the Picard group is trivial. (Similar to in algebraic number theory).
Proof: Suppose $R$ is a UFD. Then every codimension 1 prime ideal is principal. Since these generate the divisor group, every divisor is principal. Suppose $R$ is not a UFD. Take an irreducible element that is not prime. This element... | 8 | https://mathoverflow.net/users/18060 | 82832 | 49,535 |
https://mathoverflow.net/questions/82826 | 7 | Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like this:
Def. element on the group by "Euler coordinates",
$g \doteq \prod\limits\_{i=-\infty}^{\infty} e^{\omega^i X\_i}$... | https://mathoverflow.net/users/17660 | Haar measure on infinite dimensional Lie groups? | There is something called the Shavgulidze, or the Malliavin-Shavgulidze measure on Diff of smooth manifolds. You can find a discussion in Differentiable measures and the Malliavin calculus (p. 397, available on google books). It is not quite invariant, but quasi-invariant.
| 7 | https://mathoverflow.net/users/11142 | 82834 | 49,536 |
https://mathoverflow.net/questions/82836 | 0 | Every countable union of rectangles in R2 is a Lebesgue measurable set. Is the converse true, too?
Specifically, I wonder whether the following statement is true:
Let A be a set in the unit square that is Lebesgue measurable. Then there a countable collection of rectangles and a null set such that A is equal to the... | https://mathoverflow.net/users/19774 | Characterization of Measureable Sets | No, see [here](http://en.wikipedia.org/wiki/Smith-Volterra-Cantor_set) for a counter-example (it's a variant of the Cantor set which has non null Lebesgue measure and does not contain any interval).
And if you want a counter-example for $\mathbb{R}^2$ instead of $\mathbb{R}$, just cross it with an interval.
| 0 | https://mathoverflow.net/users/10217 | 82837 | 49,537 |
https://mathoverflow.net/questions/82776 | 3 | Can please someone help me with the following problem.
Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational.
Now I need to solve the inequality
$nx \; \mathrm{mod} \; 1 < v$
with respect to $n$, for some given small $v$.
According to Equidistribution Theorem, this se... | https://mathoverflow.net/users/19765 | Equidistribution Theorem: distance between solutions | Here is a quick solution to your problem. Fix a positive integer $q$ such that $\{ qx \} <v $, and then fix a positive integer $s$ such that $1<s\{qx\}$. Now assume that $n$ is any integer satisfying $\{nx\}<v$. Let $r$ be the smallest positive integer such that $1<\{nx\}+r\{qx\}$. Clearly, $r\leq s$. In addition, $\{n... | 1 | https://mathoverflow.net/users/11919 | 82846 | 49,542 |
https://mathoverflow.net/questions/82847 | 8 | I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me.
Let $F$ be a free group with $n$ generators $x\_1,\dots,x\_n$. Consider the
'augmentation' map $a:F \rightarrow \mathbb{Z}$ that sends $x\_i$ to $1$ for $i=1,\dots,n$,
and let $A = \ker f... | https://mathoverflow.net/users/9317 | A metabelian quotient of a free group | To find generators of $A$, use the Nielsen-Schreier method. It is very easy in that case: <http://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem>
| 7 | https://mathoverflow.net/users/nan | 82848 | 49,543 |
https://mathoverflow.net/questions/82827 | 4 | Let $R$ be a Dedekind ring, let $S = \mathrm{Spec} R$, and let us suppose that $f: X \to S$ is a finite morphism. Note that $X$ is not required to be connected. Does there exist a "numerical criterion" that will produce a closed subscheme $S\_0 \subset S$ such that $f$ is flat when restricted to $f^{-1}(S - S\_0)$?
... | https://mathoverflow.net/users/13410 | "Numerical Criterion" for Flatness | Take a free resolution of $X$ as an $S$-module, or, more importantly, the first two steps. Look at the map between them, which is given by a matrix over $S$. You need to compute how the rank of this matrix varies across various prime ideals, since this is a constant minus the dimension of the cokernel.
To compute the... | 2 | https://mathoverflow.net/users/18060 | 82853 | 49,546 |
https://mathoverflow.net/questions/82858 | 3 | Consider the matrix $H=H^T$, $H>0$, $H \in R^{n \times n}$, and the vector $v \in R^n$. In a numerical algorithm, I need to compute the product $b = Hv$. Right now I am following the naive approach:
$b\_i = \sum\_{j=1}^{n} h\_{ij} v\_j, i=1,...,n$.
Is there a faster way to compute this product? $H$ is non-sparse and c... | https://mathoverflow.net/users/19781 | Fast multiplication of constant symmetric positive-definite matrix and vector. | If I understand the question right, by "constant" it is meant that $H$ is a fixed, but arbitrary positive definite matrix.
In general, I don't think that you can compute the matrix-vector product $Hv$ faster than $O(n^2)$. But if $H$ has structure (Toeplitz, Circulant, Strictly diagonally dominant, etc.), or if you a... | 7 | https://mathoverflow.net/users/8430 | 82862 | 49,549 |
https://mathoverflow.net/questions/82852 | 7 | I would like an example of a J-structure $(J^A,B)$ which is not acceptable and one that is not 1-sound.
Edit:Let us recall that a structure $J^A\_\alpha$ is acceptable if for every limit ordinal $ \xi<\alpha $. $J^A\_{\xi+\omega}\models \vert \xi\vert\leq \vert \tau\vert $, whenever $\tau<\xi$ and satisfies ${\mathca... | https://mathoverflow.net/users/19778 | Acceptability and Soundness of J-structures. | For an amenable $(J,B)$ which is not 1-sound, take a non-constructible real $x$ such that $\aleph\_1^L = \aleph\_1^{L[x]}$ (and let's say $V = L[x]$ so this is the true $\aleph\_1$). Set $B = \lbrace\omega\_1+n:n \in x\rbrace$. Then $(J\_{\omega\_1+1},B)$ is amenable and $x$ is $\Sigma\_1(J\_{\omega\_1+1},B)$ (with par... | 4 | https://mathoverflow.net/users/2000 | 82865 | 49,552 |
https://mathoverflow.net/questions/82867 | 9 | Given a finitely generated subgroup of a finitely generated hyperbolic group. Is it true that the inclusion of each subgroup is a quasiisometric embedding ?
The first example for a group that does not have this property is a Baumslag-Solitar group $BS(1,m)= \langle a,b| bab^{-1}=a^m\rangle$. We have $a^{m^k}=b^kab^{-... | https://mathoverflow.net/users/3969 | Are subgroups of hyperbolic groups quasiisometrically embedded ? | The answer to the first question is "no". Look at the Rips' construction <http://en.wikipedia.org/wiki/Small_cancellation_theory>. The finitely generated normal subgroup there is usually very badly distorted. I do not see many general properties of the class of groups where every f.g. subgroup is undistorted. It contai... | 12 | https://mathoverflow.net/users/nan | 82869 | 49,553 |
https://mathoverflow.net/questions/82875 | 8 | Does there exist a set $M \subset \mathbb{R}^2$ which has the following two properties:
* Forall $x \in \mathbb{R}$ the set $\{y \in \mathbb{R} \mid (x,y) \in M\}$ is countable.
* Forall $y \in \mathbb{R}$ the set $\{x \in \mathbb{R} \mid (x,y) \notin M\}$ is countable.
| https://mathoverflow.net/users/19784 | Does there exist a subset of $\mathbb{R}^2$ which is "very small" and "very big" in the specified way? | It is a theorem of Sierpinski (Sur un theoreme equivalent a l'hypothese du continu) that the existence of such a set is equivalent to the continuum hypothesis.
| 26 | https://mathoverflow.net/users/17836 | 82877 | 49,555 |
https://mathoverflow.net/questions/82874 | 0 | Let W be a manifold with boundary such that \partial W is a union of two compact manifold A,B attached along their boundary. Does poincare duality hold for (W,A) and (W,B)?
| https://mathoverflow.net/users/6569 | A form of Lefschetz duality | Yes. Let's assume $W$ is oriented. It has a fundamental class $[W]\in H\_n(W)$, and by Lefschetz duality the cap product with $[W]$ produces isomorphisms $H^p(W,A\cup B)\to H\_{n-p}(W)$ and $H^p(W)\to H\_{n-p}(W,A\cup B)$. If you think about it, it also produces a map $H^p(W,A)\to H\_{n-p}(W,B)$. The latter is an isomo... | 3 | https://mathoverflow.net/users/6666 | 82878 | 49,556 |
https://mathoverflow.net/questions/82876 | 2 | According to "[Notes on differentiable stacks](https://www.uni-due.de/~hm0002/stacks.pdf)" by Heinloth,
>
> the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13)
>
>
>
(Here $G$ is a Lie group.) My questions are:
(1) What is the precise statement? (The category of morphisms to the cl... | https://mathoverflow.net/users/5206 | Classification of principal G-bundles over a differentiable stack | They key insight is that the bicategory of differentiable stacks is equivalent to the bicategory of Lie groupoids, where the 1-morphisms are so-called bibundles, or Hilsum-Skandalis morphisms. Under this equivalence, the statement is the following:
Let $\mathcal{X}$ be a Lie groupoid, and let $BG$ the Lie groupoid th... | 2 | https://mathoverflow.net/users/3473 | 82880 | 49,558 |
https://mathoverflow.net/questions/80706 | 9 | The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:
Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)
I would like to know if anybody knows anything concerning hist... | https://mathoverflow.net/users/19075 | History of a conjecture/problem non-inner automorphisms of order p in finite p-groups | One relevant recent direction is given in my 2007 paper with Geir Helleloid, The automorphism group of a finite p-group is almost always a p-group. J ALGEBRA vol. 312, (1) 294-329.
<http://www.sciencedirect.com/science/article/pii/S0021869307000142> also at <http://arxiv.org/abs/math/0602039>
We show that the automor... | 2 | https://mathoverflow.net/users/11043 | 82884 | 49,561 |
https://mathoverflow.net/questions/82897 | 0 | I developed the following system of two ODEs while working on a problem of copulas:
f(u) (G(u) - G(0)) = 1,
g(v) (F(1) - F(v)) = 1
Here G is a primitive of g and F is a primitive of f.
I tried to solve the system via sage, which uses maxima for this, but maxima
says it cannot solve the system. If that helps, o... | https://mathoverflow.net/users/6494 | How to solve this differential equation system? | I'll rewrite this: let $x(t) = G(t) - G(0)$ and $y(t) = F(t) - F(1)$. Then the system says
$$y'(t) x(t) = 1,\ x'(t) y(t) = -1,\ x(0)=0,\ y(1) = 0$$
However, it's obviously impossible to satisfy the differential equations at $t=0$ or $t=1$.
You say you want $u$ and $v$ to be in $(0,1)$, so maybe you could hope for
... | 9 | https://mathoverflow.net/users/13650 | 82900 | 49,566 |
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