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https://mathoverflow.net/questions/58410 | 9 | Is it true that a finitely generated subgroup of a cartesian product of free groups has a finite cohomological dimension?
The same question about pro-$p$ groups:
Is it true that a finitely generated closed subgroup of a cartesian product of free pro $p$-groups has a finite cohomological dimension?
| https://mathoverflow.net/users/10482 | Finitely generated subgroups of a product of free groups | Regarding your first question, the answer is 'yes'. Consider an arbitrary direct product of free groups $\prod\_\alpha F\_\alpha$ and $H$ a finitely generated subgroup. Then $H$ is residually free. It follows from work of Baumslag--Remeslennikov--Miasnikov (I think, originally - there are now many proofs of this fact) ... | 10 | https://mathoverflow.net/users/1463 | 58433 | 36,361 |
https://mathoverflow.net/questions/58321 | 5 | Consider two probability measures $\mu\_0$ and $\mu\_1$ on $\mathbb{R}^n$, such that $\mu\_0\ll \mu\_1$. Then I can define a "distance" like quantitiy
$$
\mathrm{Var}\_{\mu\_1}\left(\frac{\mathrm{d}\mu\_0}{\mathrm{d}\mu\_1}\right)
$$
Is this quantity already known?
For simplicity assume that both measures are absolu... | https://mathoverflow.net/users/13400 | distance in terms of the variance between two absolutely continuous probability measures | The Kullback-Leibler divergence is a special case of [Rényi divergence](http://en.wikipedia.org/wiki/R%25C3%25A9nyi_entropy#R.C3.A9nyi_divergence). In your notation, for $\alpha > 0$, the Rényi divergence of order $\alpha$ is defined by
$$
D\_\alpha(p\_0,p\_1)
= \frac{1}{\alpha - 1} \log \left( \int \left(\frac{d\mu\_... | 6 | https://mathoverflow.net/users/1044 | 58442 | 36,366 |
https://mathoverflow.net/questions/58383 | 3 | Could someone tell me the time complexity of a convex quadratically constrained quadratic program (QCQP)? Any references?
| https://mathoverflow.net/users/5531 | Complexity of convex quadratically constrained quadratic programming (QCQP) | According to the Wikipedia article at <http://en.wikipedia.org/wiki/NP-hard> it is NP-hard. The Wikipedia article gives as a reference a book which is available at <http://www.stanford.edu/~boyd/cvxbook/>
| 1 | https://mathoverflow.net/users/1098 | 58456 | 36,373 |
https://mathoverflow.net/questions/58451 | 4 | Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.
Then we have a bijection between $Ext^1\_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.
Assume all obstructions classes belonging to $F$ vanish, e.g. $Ext^2(F,F)=0$ and we have a nontrivial element $x \in Ext^1(F,F)$... | https://mathoverflow.net/users/3233 | Extension of a first order deformation of a sheaf | Let us assume that we are working over $\mathbb{C}$.
If $F$ is a *stable* sheaf then the answer to your first question is **yes**.
In fact, in this case there exists a quasi-projective moduli space $M$ and, since $Ext^2(F, F)=0$, at the point $[F]$ this moduli space is smooth, of dimension $Ext^1(F, F)$. Moreover, ... | 2 | https://mathoverflow.net/users/7460 | 58459 | 36,374 |
https://mathoverflow.net/questions/58455 | 3 | Let $p=2$ or 3, and let $k$ be an algebraically closed field of char. $p.$ Let $E$ be the supersingular elliptic curve over $k$ (with $j=0$). Let $G$ be the automorphism group of $E,$ which has order 12 (resp. 24) when $p=3$ (resp. 2). Then the $\ell$-adic cohomology $H^1(E,\mathbb Q\_{\ell})$ is a 2-dimensional repres... | https://mathoverflow.net/users/370 | supersingular elliptic curve in char. 2 or 3 | If you have a supersingular curve in general, the $\ell$-adic cohomology is a faithful representation of the endomorphism ring, which is a quaternion order that splits over $\mathbb{Q}\_\ell$. In the case of characteristic 2 and 3, the automorphism groups are non-abelian (in particular, isomorphic to $SL\_2(\mathbb{F}\... | 7 | https://mathoverflow.net/users/121 | 58463 | 36,376 |
https://mathoverflow.net/questions/58458 | 7 | A well-known mathematician once explained the following conjecture to me, as "an example of how little we know about cycles of codimension $\geq 2$." Let $C$ be a curve defined over a number field $k$, and let $x\_1,x\_2 \in C(\overline{k})$ be distinct points. Then the zero-cycle $\gamma=(x\_1,x\_1)+(x\_2,x\_2)-(x\_1,... | https://mathoverflow.net/users/1464 | Degree zero zero-cycles on the square of a curve | I think this conjecture is normally attributed to Bloch and Beilinson, and it is a special case of their general conjecture that Albanese equivalence coincides with rational equivalence (up to torsion) on smooth projective varieties over number fields.
(For varieties over any field --- of char. zero say --- the Chow gr... | 9 | https://mathoverflow.net/users/2874 | 58464 | 36,377 |
https://mathoverflow.net/questions/58440 | 4 | can any subgroup of the unitary group of full matrix alg $M\_d(\mathbb{C})$ be approximated on finite
sets by a finite subgroup?
i.e. is the following True or false?
Let $n, d$ be positive integers and let
$u\_1,..., u\_n$ be in the unitary group $U\_d=U (M\_d(\mathbb{C}))$ of $M\_d(\mathbb{C})$. Then for
every $\eps... | https://mathoverflow.net/users/13643 | Subgroups of U(M_n) | no, it is not true. the following is contained in Andreas Thom [question](https://mathoverflow.net/questions/34625/finite-subgroups-of-unitary-groups).
from the first paragraph of his question:
>
> Let $n$ be an integer. Camille Jordan
> showed that there exists some $m \in
> > {\mathbb N}$ (depending on $n$), ... | 12 | https://mathoverflow.net/users/8699 | 58473 | 36,379 |
https://mathoverflow.net/questions/58444 | 4 | Let $M$ be a submanifold in an euclidean space $\mathbb{R}^k$, and $\nu(M)$ the normal bundle to $M$, let us denote $\phi$ the restriction to $\nu(M)$ of the exponential map for $\mathbb{R}^k$.
A critical value for $\phi$ could be called focal point of $M$.
By Sard's theorem, the set of critical values for $\phi$ is ... | https://mathoverflow.net/users/12617 | Do the focal points of a submanifold $M$ in $\mathbb{R}^k$ form a closed subset? | Since the set of critical points is always closed, it suffices to assume that $\phi$ (or, more generally, its restriction to the set of its critical points) is proper. (A map is called {\it proper} if the pre-image of every compact set is compact.)
One can formulate various sufficient conditions in terms of $M$. For ... | 7 | https://mathoverflow.net/users/4354 | 58477 | 36,382 |
https://mathoverflow.net/questions/58424 | 7 | Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume that G has such a property, that is, the normal subgroups of G constitute a chain with respect to inclusion. For example, sim... | https://mathoverflow.net/users/13641 | Groups whose normal subgroups form a chain with respect to inclusion | For solvable groups without Frattini chief factors, this is equivalent to each of the following (individually):
* having a unique chief series,
* every quotient group having a faithful primitive permutation action,
* the upper Fitting series being a chief series
* the lower Fitting series being a chief series
This ... | 7 | https://mathoverflow.net/users/3710 | 58481 | 36,384 |
https://mathoverflow.net/questions/58462 | 4 | Let $F$ be a free group of finite rank $r>1$ and let $c\_1,c\_2,...$ be a full set of basic commutators for $F$. Upon completing the $k$-th step of the collection process for an arbitrary $g\in F$, one obtains a representation
$g=c\_1^{n\_1}c\_2^{n\_2}\ldots c\_{t}^{n\_t}g\_k$,
where
$c\_1,\ldots,c\_t$ are all ... | https://mathoverflow.net/users/12961 | Is the collection process for commutators potentially infinite? | Yes. I believe (ab)^3 or so, at least (ab)^n for high enough n, never terminates. As far as I recall, only (ab)^1 and (ab)^2 terminate. If you start to collect them, I think you'll see the proof.
| 4 | https://mathoverflow.net/users/3710 | 58482 | 36,385 |
https://mathoverflow.net/questions/58466 | 0 | "Call a Turing machine $A$ a *d-machine* if, for some polynomial $p(\cdot)$, when $A$ starts with any input string, say of length $n$, in its alphabet on its (otherwise blank) tape, it will halt in a number of steps bounded by $p(n)$ with its tape either blank or bearing just a string of length $n$. Then each d-machine... | https://mathoverflow.net/users/7458 | Is the following statement a correct formulation of the (much doubted) P = NP conjecture? | This is a correct formulation of P=NP, with two caveats:
* The blank character cannot be considered part of the alphabet. Otherwise a length n+1 string with a blank at the end is indistinguishable to a length n string.
* P=NP is usually defined as "If it possible to recognize a solution it is possible to find out if ... | 3 | https://mathoverflow.net/users/6279 | 58487 | 36,389 |
https://mathoverflow.net/questions/58478 | 3 | I am working on a problem where the following equation came up
$${\bf X}\_1{\bf A}{\bf X}\_2{\bf A}^T{\bf X}\_3{\bf A}-{\bf X}\_4={\bf A}$$
where ${\bf A}$ is an arbitrary $n\times n$ and ${\bf X}\_i$s are unknown diagonal real matrices.
My question is if it is feasible and if there is a computationally tractable w... | https://mathoverflow.net/users/2763 | Feasibility of a matrix equation | Looking at it entry-by-entry, you have $n^2$ equations in $4n$ variables. However, there is some symmetry here since multiplying X1, X2, X3 respectively by scalars whose product is 1 leaves the result unchanged. So there are really just $4n-2$ degrees of freedom rather than 4n. If $n \ge 4$ we have $n^2 > 4n - 2$ so I ... | 5 | https://mathoverflow.net/users/13650 | 58490 | 36,390 |
https://mathoverflow.net/questions/58493 | 5 | Is there an efficient algorithm for finding the solution $x$ of
$b = Ax$
that minimizes the Hamming weight of $x$, where
* $A$ is a nxm-matrix over the field $\mathbb{F}\_2$ ("integer matrix modulo 2") of rank $n$,
* $n<m$, say $m=500$, $n=200$,
* $b$ is a $n$-length fixed vector over $\mathbb{F}\_2$ ("a binary w... | https://mathoverflow.net/users/39843 | Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2). | No (unless P=NP). This is the decoding problem for error-correcting codes and it is known to be NP-complete.
| 10 | https://mathoverflow.net/users/2290 | 58494 | 36,391 |
https://mathoverflow.net/questions/58468 | 0 | Hello
I want to prove that
$\lim\_{h\rightarrow\infty}\left(\int\_{0}^{\infty}\left(\cos ht-1\right)\underset{t}{\triangle}\left[\frac{\phi(t)\exp\left(-itx\right)}{it}\right]dt\right)=-\int\_{0}^{\infty}\underset{t}{\triangle}\left[\frac{\phi(t)\exp\left(-itx\right)}{it}\right]dt$
where
$\underset{t}{\triangle... | https://mathoverflow.net/users/9404 | Proving uniform bound | The uniform boundedness suggested by the OP is not true, if $\phi$ is say the following function: it equals $t^2$ for $t \in [k,k + 1/k^3]$ for all $k \in \mathbb{Z}$ and $0$ elsewhere. Verify for yourself that $\phi(t)$ is still integrable, hence in $L^1$. But for the sake of Fourier inversion, all you need is that $\... | 2 | https://mathoverflow.net/users/4923 | 58501 | 36,395 |
https://mathoverflow.net/questions/58071 | 7 | Let $X$ be a compact Kahler manifold, let $D$ be a smooth divisor in $X$, and let $U$ be a tubular neighbourhood of $D$ in $X$. Suppose that $D$ is Fano. Is it possible to extend every closed (1, 1)-form on $D$ to a closed (1, 1)-form on $U$?
| https://mathoverflow.net/users/3566 | Is it always possible to extend a closed (1, 1)-form on a divisor to a closed (1, 1)-form on a tubular neighbourhood? | Yes. For the proof, see e.g. <http://arxiv.org/abs/math/0609617>
Theorem 4.1. Here a stronger result is actually proven:
Theorem: Let $(M, \omega)$ be a compact Kahler manifold,
and $Z\subset M$ a closed complex submanifold.
Denote by $[\omega]\in H^2(M)$ the Kahler class of $M$.
Consider a Kahler form $\omega\_0$ on... | 5 | https://mathoverflow.net/users/3377 | 58505 | 36,396 |
https://mathoverflow.net/questions/58414 | 6 | I am looking for an analytical solution for the linear PDE
$(1)\qquad\qquad \qquad f\_t+ A f\_x + B f = 0, $
Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector.
Clearly each one of $f\_t + A f\_x = 0,$ and $f\_t+ B f = 0,$ has an analytical solution involving eigenvectors and eigenvalues of $A$ a... | https://mathoverflow.net/users/3578 | Analytical solution to a Linear advection-reaction PDE | By *analytical*, I presume that you mean *explicit*, or *in close form*. The known case so far is when $A$ and $B$ can be diagonalized in the same basis. Notice that the case where some eigenvalues come as complex conjugate pairs is subtle.
To see the importance of the condition $[A,B]=0\_n$, let us make the Fourier ... | 7 | https://mathoverflow.net/users/8799 | 58509 | 36,398 |
https://mathoverflow.net/questions/58447 | 4 | I once thought that the analogue of bialgebras and Lie bialegras is similar to that of (associative) algebras and Lie algebras, but it seems not that trivial.
Recall the definitions: a) bialgebra $A$ is a algebra $A$ with a comultiplication $\delta: A \to A\otimes A$ such that $\delta$ is coassociative and a morphis... | https://mathoverflow.net/users/7341 | Analogue Bialgeras vs Lie bialgebras | There is of course a much more profound "analogy" between bialgebras and Lie bialgebras: the quantization and dequantization functors of Etingof and Kazhdan. From this point of view, Lie bialgebras appear as first order terms of a deformation theory which ultimately assigns to every Lie bialgebra a bialgebra by deforma... | 3 | https://mathoverflow.net/users/12482 | 58511 | 36,400 |
https://mathoverflow.net/questions/58483 | 16 | So not so long ago, I asked for a simple proof that $\mathbf{R}$ has only one smooth structure. A proof that was communicated to me by Ryan Budney ([link text](https://mathoverflow.net/questions/52620/smooth-and-analytic-structures-on-low-dimensional-euclidian-spaces)) was the following:
So let me recall his argumen... | https://mathoverflow.net/users/11765 | Looking for a simple proof that R^2 has only one smooth structure | Some comments alluded to the possibility to show this using the Riemann uniformization theorem (by paracompactness, any oriented $2$-manifold has an almost complex structure, which is integrable by Newlander-Nirenberg and by the uniformization theorem, it will be biholomorphically equivalent to the plane or the unit di... | 14 | https://mathoverflow.net/users/9928 | 58515 | 36,404 |
https://mathoverflow.net/questions/58527 | 5 | A classification (up to conjugacy) of all closed subgroups of the (identity component of the) Lorentz group $\mathrm{SO}(1,3)\_0$ in terms of the subalgebras of its Lie algebra was given in
* R. Shaw. *The subgroup structure of the homogeneous Lorentz group*. The Quaterly Journal of Mathematics, Oxford **21** (1970) ... | https://mathoverflow.net/users/8278 | Subgroup structure of $\mathrm{SO}(1,n)_0$ | I would expect this to become impractical (although in practice there is an algorithm) as $n$ increases. There are results for $n=4$ in this paper:
*Quantum numbers for particles in de Sitter space* by J Patera, P Winternitz, H Zassenhaus.
Published in the [Journal of Mathematical Physics (1976) vol. 17 (5) pp. 717-7... | 3 | https://mathoverflow.net/users/394 | 58529 | 36,413 |
https://mathoverflow.net/questions/58497 | 98 | My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets and not just plain sets; to make $\mathbb{A}^1$-homotopy work, you work with simplicial (pre?)sheaves and not just plain... | https://mathoverflow.net/users/344 | Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"? | I don't think I have a compelling answer to this question, but maybe some bits and pieces that will be helpful. One point is that all of the examples that you bring up are related to the first: simplicial sets can be used as a model for the homotopy theory of spaces. Pretty much any homotopy theory can be "described" i... | 103 | https://mathoverflow.net/users/7721 | 58530 | 36,414 |
https://mathoverflow.net/questions/58534 | 2 | Hi,
I have the following situation: $R,H$ schemes (can be assumed noetherian and of finite type) over a field $k$ which we can assume to be algebraically closed, with $H$ reduced, $Y\subset R\times \mathbb{P}\_k^n$ an open subset, $p:Y\rightarrow R$ the restriction of the projection onto the first factor and $w:Y\rig... | https://mathoverflow.net/users/12198 | formal smoothness versus reducedness | Since $H$ is reduced and $Y$ is smooth over $H$ (I am assuming that everything is finite type over $k$, so smooth and formally smooth are the same) we see that $Y$ is reduced.
So the problem is the following: show that if $Y \subset R \times \mathbb P^n$ is open and reduced, and the projection $Y \to R$ is surjective... | 6 | https://mathoverflow.net/users/2874 | 58544 | 36,419 |
https://mathoverflow.net/questions/58543 | 2 | I learned analysis a while ago, so let me define what I want. Suppose we have a set whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that this set is (Lebesgue) measurable? What if we also suppose that the tangents change continuously? I am interested in any modification of th... | https://mathoverflow.net/users/955 | Are there non-measurable sets with smooth boundary? | As was pointed out, the question did not intend the set to be the interior of the curve. The question now boils down to whether a differentiable Jordan curve has measure zero. This follows from the Lebesgue density theorem, see e.g. <http://en.wikipedia.org/wiki/Lebesgue>'s\_density\_theorem.
If the curve is differen... | 4 | https://mathoverflow.net/users/12120 | 58545 | 36,420 |
https://mathoverflow.net/questions/47818 | 22 | When Grothendieck and his followers were working on their profound progress of algebraic geometry, did they ever consider non-commutative rings? Is there anyway evidence that Grothendieck foresaw the developments that would later come in non-commutative geometry or quantum group theory?
| https://mathoverflow.net/users/1867 | Grothendieck and Non-commutative Geometry? | No and yes, depending on the level of understanding. The consideration of noncommutative rings telling about geometry is almost nonexistent in Grothendieck's published opus. One of the exceptions is that he considered cohomologies for the possibly noncommutative sheaves of $\mathcal{O}$-algebras for commutative $\mathc... | 29 | https://mathoverflow.net/users/35833 | 58549 | 36,423 |
https://mathoverflow.net/questions/58425 | 2 | Let $S\_n$ denote the set of partions of $n$ such that every part is greater than 1. Partitions $(x\_1,\ldots,x\_k), (y\_1,\ldots,y\_l) \in S\_n$ are said to have almost equal product if $$\prod\_{i=1}^k (x\_i+1) = \prod\_{i=1}^l (y\_i+1)$$.
For example if $n = 14$ the partitions (3,3,8) and (2,5,7) are almost equal ... | https://mathoverflow.net/users/1737 | Number of partitions of $n$ with different product | I don't know what you mean by "at least subexponential." A subexponential upper bound is trivial because the number of partitions of $n$ with no restrictions grows subexponentially.
A little work gives a superpolynomial lower bound. Choose an arbitrary subset $\{p\_1,...p\_k\}$ of the primes between $5$ and $f(n)$ f... | 4 | https://mathoverflow.net/users/2954 | 58552 | 36,425 |
https://mathoverflow.net/questions/58559 | 19 | I'm trying to understand the dualizing sheaf $\omega\_C$ on a nodal curve $C$, in particular why is $H^1(C,\omega\_C)=k$, where $k$ is the algebraically closed ground field. I know this sheaf is defined as the push-forward of the sheaf of rational differentials on the normalization $\tilde{C}$ of $C$ with at most simpl... | https://mathoverflow.net/users/13139 | dualizing sheaf of a nodal curve | If $\tilde{C}$ is the normalization, with two points $x$ and $y$ being identified under the map $\pi: \tilde{C} \to C$ to the node $z$ of $C$, then we have an exact sequence
$$0 \to \Omega^1\_{\tilde C} \to \Omega^1\_{\tilde C}(x + y) \to k\_x \oplus k\_y \to 0,$$
where $k\_x$ and $k\_y$ are the skyscraper sheaves at ... | 27 | https://mathoverflow.net/users/2874 | 58563 | 36,431 |
https://mathoverflow.net/questions/58541 | 6 | Hello,
I try to understand aspects of homotopy coherence, in particular "recognition principle" of May.
About the following I did not think a lot, but I decided to ask here anyway, so to save reinvention of the wheel and get clarifying comments.
Consider $E\_1$ - the topological operad of small $1$-cubes. An $E\_1$... | https://mathoverflow.net/users/2095 | Classifying spaces of E_1 - spaces | From the horse's mouth.
1. I would think a good theory of parametrized $E\_1$-spaces should not be too
hard to develop, along the general lines of parametrized spaces (and spectra)
as developed ad nauseum in
J.P. May and J. Sigurdsson. Parametrized homotopy theory.
Presumably the fibers should be grouplike. A c... | 14 | https://mathoverflow.net/users/14447 | 58568 | 36,432 |
https://mathoverflow.net/questions/58583 | 7 | Suppose L is a partial differential operator of arbitrary order with constant coefficients.
If u is in $L^p(\mathbb{R}^n)$ and Lu=0 in distributions, is it necessarily the case that u=0? Does the answer depend on p?
Also, if u is a compactly supported distribution in $\mathbb{R}^n$ with Lu=0 (in the usual sense, i... | https://mathoverflow.net/users/13678 | Uniqueness of weak solution L[u]=0 | If $Lu=0$, then the Fourier transform of $u$ must have its support on the manifold where the symbol of $L$ is zero. Hence the Fourier transform of $u$ cannot be a function. This rules out $u\in L^p$ for $p\le 2$; it also rules out a compactly supported distribution. The Bessel function $J\_0(\sqrt{x^2+y^2})$ satisfies ... | 3 | https://mathoverflow.net/users/12120 | 58588 | 36,442 |
https://mathoverflow.net/questions/58591 | 4 | A friend in physics asked this question, and I didn't know the answer.
Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative curvature?
| https://mathoverflow.net/users/35508 | Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature | I guess you mean constant curvature $=-1$; otherwise you get an example by rescaling.
On a sphere with two handles there are metrics with curvature $\equiv-1$ which look roughly as a long neck attached to two tori.
Such metric has as small eigenvalue as you want.
On the other hand upper diameter + lower curvature ... | 4 | https://mathoverflow.net/users/1441 | 58593 | 36,445 |
https://mathoverflow.net/questions/58577 | 8 | I've been doing some light(?) reading on motives and the standard conjectures in an attempt to put various things that I tangentially know in perspective.
The question is this: the Weil conjectures assert that $Z=\frac{P\_1(t)...P\_{2r-1}(t)}{P\_0(t)...P\_{2r}(t)}$ where the $P\_i$'s are certain polynomials. (the ass... | https://mathoverflow.net/users/5309 | Why is the zeta function of a variety over a finite field not a polynomial? (question about motives) | The zeta function of a variety $X$ over a finite field is *a priori* defined to be a point counting function, i.e. it is the following product over the closed points of $X$ (thought of as a scheme):
$$\zeta\_X(s) = \prod\_{x}(1 - | \kappa(x)|^{-s})^{-1},$$
where $\kappa(x)$ is the residue field of $x$ and $|\kappa(x)|$... | 23 | https://mathoverflow.net/users/2874 | 58598 | 36,449 |
https://mathoverflow.net/questions/58601 | 7 | I have been looking at this for days and I am going insane.
I need to show that for a Dirichlet series equal to $\zeta(s)\zeta(2s)$, the sum of the coefficients less than $x$ is $x\zeta(2)+O(x^{(3/4)})$, and then expand that to the $\Pi \zeta(ks)$ for all $k$ in an effort to find the formula for the number of non-is... | https://mathoverflow.net/users/13623 | Contour integration of $\zeta(s)\zeta(2s)$ | This case, at least, you can do by hand; if $c(n)$ is the $n$th coefficient of $\zeta(s)\zeta(2s)$, then
$\sum\_{n\leq X}c(n)=\sum\_{nm^2\leq X}1=\sum\_{m}\sum\_{n\leq X/m^2}1 = \sum\_{m < \sqrt{X}}(m^{-2}
X+O(1))=\zeta(2)X+O(\sqrt{X}).$
| 7 | https://mathoverflow.net/users/1464 | 58604 | 36,450 |
https://mathoverflow.net/questions/58600 | 12 | I pick a random subset $S\subseteq\lbrace1,\ldots,N\rbrace$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need to determine the subset? (If there is only one possibility left, then you can omit the last guess.)
There is an obvious str... | https://mathoverflow.net/users/4758 | Guessing a subset of {1,...,N} | This is a well-studied problem, sometimes phrased as a coin-weighing problem. It is known that $g(N)$ is $O(N / \log N)$. (We can even specify the guessing sets in advance, without knowing the previous answers.) I believe these three papers are the earliest to show this bound:
B. Lindstrom (1964), "On a combinatory d... | 19 | https://mathoverflow.net/users/3376 | 58607 | 36,452 |
https://mathoverflow.net/questions/58567 | 2 | For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n\_m]$. Write $J \equiv (j\_1,\dots,j\_N) \in \prod\_m [n\_m]$ with $j\_m \in [n\_m]$.
The composite Markov generator corresponding to running each of the $N$ processes i... | https://mathoverflow.net/users/1847 | Spectral gap of a product of Markov processes | The spectral gap is just the smallest of the spectral gaps of the component chains. The matrix $Q$ can be simply written as the sum of $Q^{(1)}\otimes I\otimes\ldots\otimes I$, $I\otimes Q^{(2)}\otimes I\otimes\cdots\otimes I$ etc.
If the matrix $Q^{(i)}$ has eigenvalues $(\lambda^{(i)}\_j)\_{j=1}^{n\_i}$, then the ma... | 1 | https://mathoverflow.net/users/11054 | 58610 | 36,454 |
https://mathoverflow.net/questions/58589 | 9 | I was reading about [Calabi-Yau manifolds](http://en.wikipedia.org/wiki/Calabi-Yau_manifold), about which I know little, and was wondering
if these (or related complex manifolds, perhaps [K3 surfaces](http://en.wikipedia.org/wiki/K3_manifold)) can be viewed as configuration
spaces (or [moduli spaces](http://en.wikipedi... | https://mathoverflow.net/users/6094 | Calabi-Yau manifolds and polygonal linkage configuration spaces: related? | An illustrative example: *the moduli space $M$ of regular pentagons with edges of unit length*. This embeds as an open, dense subset of a compact complex surface $\bar{M}$ with a canonical Kaehler form. This surface, a 4-fold blow-up of $\mathbb{CP}^2$, is not Calabi-Yau (trivial canonical bundle) but Fano (ample antic... | 9 | https://mathoverflow.net/users/2356 | 58612 | 36,455 |
https://mathoverflow.net/questions/58533 | 0 | Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap\_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.
**Question**: Can we say that $F$ is bounded in $L^q(0,T;C^2(\bar{\Omega}))$ for any $q\in [1,\infty)$?
| https://mathoverflow.net/users/8901 | Is this (interpolation) inequality right? | This is **not an answer**, but just a formulation of a simpler analogous problem, which might help better understand the original question. Let $I=[0,1]$ and $Q=I\times I$ . Let $ 1 < p < +\infty$ , and let $\langle M\_k\rangle\in[1,+\infty[\ ^{\mathbb N}$ satisfy $M\_k +1 < M\_{k+1}$ for all $k\in\mathbb N$ . Let $\|u... | 2 | https://mathoverflow.net/users/12643 | 58628 | 36,467 |
https://mathoverflow.net/questions/58638 | 1 | Suppose we have an open set S whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that for every epsilon there is a P polygon contained in S such that there is a (1+epsilon) scaled copy of P that contains S?
| https://mathoverflow.net/users/955 | Is there a good approximating polygon for every smooth set? | I do not see how scaling could give such a property. Consider an annulus in the plane (remove a part of it to make it a Jordan domain and also make it smooth if you want). Scaling any polygon inside the annulus by $1 + \epsilon$ makes also the "missing ball" inside the annulus larger.
| 7 | https://mathoverflow.net/users/11716 | 58642 | 36,475 |
https://mathoverflow.net/questions/58633 | 18 | (Previously posted on [math.SE](https://math.stackexchange.com/questions/26909/does-every-irreducible-representation-of-a-compact-group-occur-in-tensor-products) with no answers.)
Let $G$ be a compact Lie group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. Is it true that every (c... | https://mathoverflow.net/users/290 | Does every irreducible representation of a compact group occur in tensor products of a faithful representation and its dual? | You can mimic the standard finite group argument. Recall that, if $X$ and $Y$ are two finite-dimensional representations of $G$, with characters $\chi\_X$ and $\chi\_Y$, then $\dim \mathrm{Hom}\_G(X,Y) = \int\_G \overline{\chi\_X} \chi\_Y$, where the integral is with respect to Haar measure normalized so that $\int\_G ... | 22 | https://mathoverflow.net/users/297 | 58644 | 36,476 |
https://mathoverflow.net/questions/58643 | -1 | Is it true that when the first fundamental group of a topological space $X$ is isomorphic to $\mathbb{Z}$ then $X$ is homeomorphic to $S^1 \times Y$ where the first fundamental group of $Y$ is trivial?
With a discussion with my of friends, the above question turned into (!) finding a topological space $X$ s.t. there ... | https://mathoverflow.net/users/13351 | Is $X$ homeomorphic to $S^1 \times Y$? | The M"obius band is a counterexample (to the original and current versions of the stated question).
| 2 | https://mathoverflow.net/users/6794 | 58645 | 36,477 |
https://mathoverflow.net/questions/58640 | 2 | Let X be a Hausdorff k-space (Hausdorff compactly generated space) and h a bijection on X such that for any subset E of X we have
```
E compact <=> h(E) compact.
```
Question: Does it follow that h is a homeomorphism? (The converse is true for any space X since the continuous image of a compactum is compact).
... | https://mathoverflow.net/users/13692 | homeomorphisms on k-spaces | Yes. If you restrict $h$ to any compact subset $E$, then $h$ gives a homeomorphism from $E$ to $h(E)$, because a subset of $E$ (or $h(E)$) is closed iff it is compact, so $h$ and its inverse both preserve closed sets. By compact generation, this implies that both $h$ and its inverse are continuous, so $h$ is a homeomor... | 4 | https://mathoverflow.net/users/75 | 58650 | 36,478 |
https://mathoverflow.net/questions/58646 | 2 | Hello,
in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and which behave very similarly. These include
* $\mathbb R^n$ with a $p$-norm
* $l^p$ spaces
* $L^p$ spaces
* $p$-Trace-norm... | https://mathoverflow.net/users/2082 | General theory for p-normed spaces | As Nate and I pointed out in comments, your question reduces to asking whether there is a unified framework which includes both $L^p$ spaces and Schatten spaces. One such framework (there may be others) is **noncommutative $L^p$ spaces**. (Usual $L^p$ spaces are the commutative special case.) There's a [nice survey art... | 7 | https://mathoverflow.net/users/1044 | 58651 | 36,479 |
https://mathoverflow.net/questions/49160 | 4 | This question is related to my earlier question
[here](http://tiny.cc/ga9k1) .
Given an $n\times n$ random matrix $A$, is determining the properties (mean, variance,moments,etc.) of its induced $p$-norm ($p\neq 0,1,2,\infty$) a good research problem?
Looking at related work, I couldn't find anything much except a... | https://mathoverflow.net/users/6495 | Induced p-norm of a Random matrix | Here are some papers you might want to look at if you're interested in this issue (probably better than the paper of mine that Suvrit mentioned):
1. [G. Bennett, V. Goodman, and C.M. Newman. Norms of random matrices.
Pacific J. Math. 59 (1975), no. 2, 359–365.](http://projecteuclid.org/euclid.pjm/1102905342)
2. Graha... | 3 | https://mathoverflow.net/users/1044 | 58666 | 36,488 |
https://mathoverflow.net/questions/58656 | 6 | As I understand it, there is a subfactor whose principal graph is the affine Dynkin diagram $\tilde{A}\_n$. Since every vertex has two neighbors, does that mean the space of 1-boxes is two dimensional? Is that allowed?
| https://mathoverflow.net/users/3046 | What is the subfactor planar algebra of type $\tilde{A}_n$, of index 4? | One concrete realization of this planar algebra goes as follows.
* Start with the (generalized) PA freely generated by an *oriented* strand.
* Impose "$Z$-homology" relations: (a) oriented saddle moves, and (b) erase small loops (loop value = 1).
* Now introduce an unoriented strand which is the formal direct sum of... | 6 | https://mathoverflow.net/users/284 | 58668 | 36,489 |
https://mathoverflow.net/questions/58648 | 4 |
>
> **Question.** Is the $F$-polynomial of an indecomposable quiver representation irreducible?
>
>
>
Here the $F$-polynomial is the generating function of the Euler characteristics of quiver Grassmannians, that is,
$$F\_M =\sum\_{e\_1,...e\_n} \chi( \mathrm{Gr}\_{e\_1,...e\_n} M) x\_1^{e\_1} ... x\_n^{e\_n}$... | https://mathoverflow.net/users/13693 | Irreducibility of the F-polynomial of an indecomposable quiver representation | No.
Consider the Kronecker quiver: two vertices with two arrows between them. Consider the representation with dimension vector $(3,3)$ given by the matrices
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad
\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}.$$
The correspon... | 5 | https://mathoverflow.net/users/297 | 58677 | 36,493 |
https://mathoverflow.net/questions/43614 | 9 | Is it true that any complete metric of nonnegative (Gauss) curvature on $\mathbb R^2$ is conformally equivalent to the standard Euclidean metric on $\mathbb R^2$?
**Remarks:** locally any Riemannian metric on a surface is conformally equivalent
to the standard $\mathbb R^2$, due to existence of isothermal coordinate... | https://mathoverflow.net/users/1573 | Is every nonnegatively curved plane conformal to the complex plane? | Cheng-Yau proved that: A complete Riemannian manifold with non-negative Ricci curvature
and at most quadratic growth for volumes of balls as the radius goes to infinity is
parabolic.
EDIT (by Igor Belegradek). Various criteria for parabolicity are found in the [survey](http://www.ams.org/journals/bull/1999-36-02/S027... | 7 | https://mathoverflow.net/users/4696 | 58679 | 36,495 |
https://mathoverflow.net/questions/58389 | 7 | This is a general question for the homotopy theory crowd: How does one go about computing the homology and homotopy groups of homotopy fixed point spaces $X^{hG}:= Map^G(EG, X)$ for the action of a group $G$ on a space $X$? There seem to be some tools:
1. Lannes' theory: which allows you to compute (or at least say *... | https://mathoverflow.net/users/4649 | Homology of homotopy fixed point spaces | Hej Craig,
Re (2) as Tilman says in his comment, there is an unstable homotopy fixed point spectral sequence, a special case of the spectral sequence of a homotopy limit as described by Bousfield and others.
Re (1) when X is finite (and more generally), Lannes theory should be seen as generalization of ordinary Smi... | 4 | https://mathoverflow.net/users/6574 | 58689 | 36,498 |
https://mathoverflow.net/questions/58673 | 6 | For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental groups.
What about complete smooth curves, or more generally higher dimensional varieties? Are there results or conjec... | https://mathoverflow.net/users/370 | finite quotients of fundamental groups in positive characteristic | For a supersingular elliptic $E$ over an algebraically closed field of characteristic two or three there exists a smooth curve $C$ of higher genus such that $Aut\_0(E)$ is a finite quotient of $\pi\_1(C)$.
This is explained in section 3 of <http://arxiv.org/abs/1005.2142v3>
This is an easy application of a general ... | 2 | https://mathoverflow.net/users/5273 | 58691 | 36,499 |
https://mathoverflow.net/questions/58690 | 31 | Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)\subset O$, where $K$ is any compact subset of $X$, and $O$ is any open subset of $Y$. So a basis of open sets is given... | https://mathoverflow.net/users/13700 | Compact open topology on $\mathrm{Homeo}(X)$ | The following article gives you a lot of information on the question you are asking:
On Homeomorphism Groups and the Compact-Open Topology,
Jan J. Dijkstra
<http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf>
<http://www.jstor.org/pss/30037630>
The answer is in general "no".
| 21 | https://mathoverflow.net/users/3676 | 58694 | 36,501 |
https://mathoverflow.net/questions/58696 | 140 | I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics and physics. I visited a course on Lie groups, and an elementary one on Lie algebras. But I don't fully understand how th... | https://mathoverflow.net/users/13700 | Why study Lie algebras? | Here is a brief answer: Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. Most, if not all, of differential geometry centers around this. By differentiating the Lie group action, you get a Lie algebra action, which is a linearization of the group action. As a li... | 136 | https://mathoverflow.net/users/613 | 58700 | 36,502 |
https://mathoverflow.net/questions/58698 | 1 | Can anyone tell me why it is that Lie groups seem to have their second fundamental group $\pi\_2(G)$ equal to $0$, or provide me with a link to an article or a book reference?
I came across this fact reading an article where the author considers principal $G$ bundles with $G$ a simply connected simple group.
thank ... | https://mathoverflow.net/users/13700 | nullity of the second fundamental group of a Lie group | See: [Homotopy groups of Lie groups](https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups)
| 3 | https://mathoverflow.net/users/7311 | 58705 | 36,504 |
https://mathoverflow.net/questions/58697 | 18 | The following problem cropped up whilst considering generalised quadrangles with a product structure, and it boils down to a simple number theoretic problem. Let $s$ be an integer greater than 2 and suppose the geometric series $(s^r-1)/(s-1)$ is a nontrivial power of a positive integer. It seems the following is true:... | https://mathoverflow.net/users/5043 | A geometric series equalling a power of an integer | This is a well-investigated Diophantine equation known as Nagell--Ljunggren equation (they investigated this equation in the 1920s and 1940s, resp). Indeed, it is conjectured that the three solutions mentioned by the questioner are the only ones; however, it is not even known that the number of solutions is finite, tho... | 32 | https://mathoverflow.net/users/nan | 58715 | 36,509 |
https://mathoverflow.net/questions/58569 | 6 | The following question came up in my arithmetic geometry course yesterday. Suppose $\alpha$ is an irrational real algebraic integer, and suppose $\epsilon >0$ is given. Then by Roth's theorem there are at most finitely many rational numbers $\frac{h}{q}$ with $\gcd(h,q)=1$, $q>1$, such that
$$
\left| \alpha - \frac{h}... | https://mathoverflow.net/users/10458 | Question related to Diophantine approximations and Roth's theorem | My other answer was for the first version of this question. The question has now been changed completely. As Antoine mentioned in his comment, an effective Roth's theorem is not known in general. Finding such a result is the probably the main open problem in Diophantine approximation. There some instances in which a no... | 3 | https://mathoverflow.net/users/2290 | 58723 | 36,513 |
https://mathoverflow.net/questions/58707 | 15 | Consider $J^{g-1}$, the variety of degree $g-1$ line bundles on a compact Riemann surface of genus $g$. Recall that $J^{g-1}$ is a torsor for the Jacobian, thus has dimension $g$. We can produce elements in $J^{g-1}$ by choosing $g-1$ points $p\_1,\ldots,p\_{g-1}$ and constructing the associated line bundle to the divi... | https://mathoverflow.net/users/1116 | Construction of the determinant line bundle on the degree $g-1$ Picard variety | Let's say that your (compact!) Riemann surface is $X$ (of genus at least $1$). What does it mean to say that $J^{g-1}$ is the "variety of degree $g-1$ line bundles?" One way to formalize this is to say that there is a line bundle $\mathcal{L}$ on the product $X \times J^{g-1}$ with the property the rule $p \mapsto \mat... | 16 | https://mathoverflow.net/users/5337 | 58728 | 36,516 |
https://mathoverflow.net/questions/58724 | 1 | It is an elementary fact that when the number of variables exceeds the number of linear equations then the system has a nontrivial solution.
I want to know whether there is such thing about homogeneous polynomials of degree $2$ or $3$, that is, for every natural number $n$ there exists $N \in \mathbb{N}$ s.t. for eve... | https://mathoverflow.net/users/13351 | homogeneous polynomials with suitable number of variables that has a non-trivial solution | 1) It is not entirely true what you wrote about linear equations. If your equations are contradictory, then no matter how many additional variables you add, you will not find a solution. For instance $x+y=0$, $x+2y=1$, $2x+y=1$ has no solution regardless possible additional variables.
2) If you mean *homogenous* equa... | 5 | https://mathoverflow.net/users/10076 | 58729 | 36,517 |
https://mathoverflow.net/questions/58688 | 2 | Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the two closed half-spaces bounded by $H$, we call $H \cap C$ a $j$-face of $C$, where $j$ is the affine dimension of $H \ca... | https://mathoverflow.net/users/12722 | Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map | Yes this is always true. One just needs to exhibit enough polyhedra so that the span of their f-vectors is $n-1$ dimensional. Such a family is given by the polyhedra $\Delta^k\times I^{n-k}$, where $\Delta^k$ is the $k$-simplex and $I^{k}$ is the $k$-cube.
The same argument can be used that the corresponding dimensio... | 4 | https://mathoverflow.net/users/2384 | 58731 | 36,519 |
https://mathoverflow.net/questions/58739 | 21 | Hi everybody! I am reading some papers about Casson's invariant for (integral) homology 3-spheres...as the wiki says "Informally speaking, the Casson invariant counts the number of conjugacy classes of representations of the fundamental group of a homology 3-sphere M into the group $SU(2)$". It seems to be something in... | https://mathoverflow.net/users/13715 | Why is Casson's invariant worth studying? | Wikipedia's description of the Casson invariant gives the first important reason to study it. As an invariant that comes from the $\text{SU}(2)$ representation variety of $\pi\_1(M)$, it reveals in particular that $\pi\_1(M)$ is non-zero. At the time, before Perelman's proof of the Poincaré conjecture and geometrizatio... | 32 | https://mathoverflow.net/users/1450 | 58741 | 36,525 |
https://mathoverflow.net/questions/58718 | 4 | In the middle of page 9 of
<http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf>.
They said " Now we select a random subset....choosing lines independently with
probability $\frac{Q}{100}$. With positive probability....
I can not see why there is positive probability...
Could any one explain a bit about wha... | https://mathoverflow.net/users/13289 | Degree reduction argument in Guth-Katz'sproof of Erdos distinct distance problem in the plane | Not a complete answer but a quick explanation of how I read page 9 in that paper.
1. The underlying probabilistic model is that for every line $l\in\mathfrak L'$ you throw a coin that shows head with probability $100/Q$ and tail with probability $(Q-100)/Q$, and then $\mathfrak L''$ is the set of all lines whose coin... | 3 | https://mathoverflow.net/users/12674 | 58742 | 36,526 |
https://mathoverflow.net/questions/58711 | 3 | Consider a function $f$ on $S^2,$ and its spherical transform $\hat{f}.$ Let $r$ be a rotation by some $\rho \in SO(3).$ Is there some nice formula for $\widehat{f \circ r}?$ I have found some allusions to "Wigner D-matrices"... I am sure there are good references...
| https://mathoverflow.net/users/11142 | Spherical transforms and rotations | Suppose your function is $f(s)$ and its spherical transform, as you call it, is $\hat{f}\_{lm}=\int\_{S^2} Y\_{lm}(s)^\* f(s)\,\mathrm{d}s$, where the $Y\_{lm}(S)$ are spherical harmonics. Then for fixed $l$, the components of $\hat{f}$ will transform under a rotation $r\in SO(3)$ in an irreducible representation of $S... | 3 | https://mathoverflow.net/users/2622 | 58744 | 36,527 |
https://mathoverflow.net/questions/58489 | 4 | Assume that $(A,m)$ is a Noetherian normal local domain, $K = Quot(A) \subset E, F$ Galois extensions of $K$. If $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$ and we choose primes $m\_B, m\_C$, and $m\_D$ (in the corresponding rings) over $m$, then is it true that the separable part of residue fie... | https://mathoverflow.net/users/13654 | Maximal separable extensions of residue fields | Counterexample based on Hagen's remark: take $A=\mathbb{R}[[t]]$, $B=\mathbb{R}[[\sqrt{t}]]$, $C=\mathbb{R}[[\sqrt{-t}]]$. Both have residue field $\mathbb{R}$, but $D$ contains a square root of $-1$ (namely $\sqrt{-t}/\sqrt{t}$), so its residue field must be $\mathbb{C}$.
| 1 | https://mathoverflow.net/users/7666 | 58745 | 36,528 |
https://mathoverflow.net/questions/58743 | 0 | I'm implementing arithmetics for elliptic curves over secp256r1 as a homework assignment.
For scalar multiplication, the assignment specifically specifies that $k$ is "any hexidecimal encoded integer" (for the multiplication $kP$).
How is multiplication defined for negative values of $k$? My best guess is that $k$ ... | https://mathoverflow.net/users/13716 | Elliptic curve over finite field: scalar multiplication | I am not familiar with some of the abbreviations you use, and do not know what is better in your case, but:
a. for any commutative group $(G,+)$ one has $kP = (-k) (-P)$, k integer and $P$ in $G$. So, computing one inverse in $G$ (and one in the integers) one can reduce to scalar multiplication for positive integers... | 2 | https://mathoverflow.net/users/nan | 58748 | 36,529 |
https://mathoverflow.net/questions/58746 | 1 | Good Morning,
I've been trying to brush up a bit on linear systems lately, and I've ran into the following (seemingly) contradictory statements. Hopefully someone can tell me where I'm going wrong here (and say how to go right).
Consider $H^0(\mathbb{P}^1, \mathcal{O}(3))$ which is the $k$ vector space spanned by d... | https://mathoverflow.net/users/13717 | Basic Question concerning linear systems | The sections $s\_1=x^3,s\_2=x^2y,s\_3=xy^2$ and $t\_1=x^2,t\_2=xy,t\_3=y^2$ are not the same sections, in fact they correspond to different line bundles. The $s\_i$ do have a base-point, and as you remark, they give the same map as the $t\_i$ outside the base-locus.
And see this meta thread for how to merge your use... | 2 | https://mathoverflow.net/users/3996 | 58749 | 36,530 |
https://mathoverflow.net/questions/58622 | 6 | Let $X$ be the intersection of two quadrics in $P^5$. It is well known that the intermediate Jacobian $J(X)$ is isomorphic to $J(C)$ for a genus 2 curve, related to the pencil of quadrics whose base locus is $X$.
It seemed then natural to me to ask the following question:
Is there an explicit construction where $X$... | https://mathoverflow.net/users/4096 | Fano 3-fold of degree 4 | The projection from a line $L\_0$ is a birational isomorphism of $X$ onto $P^3$. It decomposes as the blow-up of the line $L\_0$ followed by the contraction of a surface swept by lines intersecting $L\_0$ onto a curve of genus 2 in $P^3$.
| 5 | https://mathoverflow.net/users/4428 | 58756 | 36,533 |
https://mathoverflow.net/questions/58752 | 6 | The existence of a countable approximate unit in a $C^{\*}$-algebra $B$ is equivalent to the existence of a strictly positive element $h\in B$. There are several ways to construct an approximate unit from $h$. My question is, does $h(h+\frac{1}{n})^{-1}$ constitute an approximate unit for $B$?
| https://mathoverflow.net/users/13718 | Approximate units from strictly positive elements in $C^{*}$-algebras. | I think this works: Functional calculus shows that $h h (h+1/n)^{-1} \rightarrow h$. Then $h$ is strictly positive if and only if $Bh$ is dense in $B$ (See, for example, Jensen+Thomsen, "Elements of KK-Theory", Lemma 1.1.21). So for $b\in B$ and $\epsilon>0$, we can find $c\in B$ with $\|b-ch\|<\epsilon$, and for all $... | 7 | https://mathoverflow.net/users/406 | 58763 | 36,538 |
https://mathoverflow.net/questions/58755 | 1 | As a by-product of a numerical linear algebra result on structured matrices, I can prove the following result:
>
> For each $m$-dimensional subspace $\mathcal{U}$ of $\mathbb{C}^n$, one can find a $n\times n$ permutation matrix $\Pi$ such that $\mathcal{U}=\operatorname{span} \Pi\begin{bmatrix}I\\\\X\end{bmatrix}$ ... | https://mathoverflow.net/users/1898 | "best" local chart for an element of $Gr(n,2n)$ | I can't give you a reference, but I can give you a quick proof. There is nothing special about $Gn(n,2n)$, so I'll prove it for $Gr(k,m)$.
Consider the $\binom{m}{k}$ Plucker coordinates. One of them must be the largest; permute basis coordiantes such that $|p\_{12\cdots k}| \geq |p\_I|$ for every $I$. Work in the c... | 3 | https://mathoverflow.net/users/297 | 58766 | 36,540 |
https://mathoverflow.net/questions/58747 | 5 | Let $A$ be a $C^\*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary group in the multiplier algebra $M(A \otimes \mathbb{K})$ is contractible in the norm topology. It was then shown by Troit... | https://mathoverflow.net/users/3995 | Is the unitary group of $l^2(A)$ with the strict topology contractible? | I would post this as a comment but as it just happens I can't do that. I do think that
the exercise that you mention proves strict contractibility. The same formula for the homotopy,
$$
(u,t)\mapsto w\_tuw\_t^\*+(1-w\_tw\_t^\*),
$$
is given in Proposition 12.2.2 of Blackadar's book on K-theory, although the statement... | 4 | https://mathoverflow.net/users/13381 | 58772 | 36,543 |
https://mathoverflow.net/questions/58681 | 19 | Let $G$ be a group and let $H$ be a subgroup of finite index.
Let $V$ be an irreducible complex representation of $G$ (no topology or anything: $V$ is just a non-zero complex vector space with a linear action of $G$ and no non-trivial invariant subs).
Now consider $V$ as a representation of $H$. Is $V$ a finite dir... | https://mathoverflow.net/users/1384 | Clifford theory: behaviour of a very general irreducible representation under restriction to a finite index subgroup. | Since $V$ is irreducible, it is a finitely generated $\mathbb C[G]$-module, any non-zero element is a generator. Since $H$ is of finite index, $\mathbb C[G]$ is a finitely generated
$\mathbb C[H]$-module. Hence $V$ is a finitely generated $\mathbb C[H]$-module. Zorn's Lemma implies the existence of an irreducible quot... | 9 | https://mathoverflow.net/users/13024 | 58773 | 36,544 |
https://mathoverflow.net/questions/58751 | 12 | Let $B$ be a $C^{\*}$-algebra and $\mathcal{B}$ a dense \*-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a Banach algebra in a norm $\|\cdot\|\_{1},$ satisfying
$\|\cdot\|\leq\|\cdot\|\_{1}$.
Also, there is a countable boun... | https://mathoverflow.net/users/13718 | Ideals in smooth subalgebras of C*-algebras | $\newcommand{\norm}[1]{\Vert#1\Vert}$
In general, I think the answer to your question is no. Take ${\mathcal B}=C^1[-1,1]$ with the norm
$\norm{f}= \norm{f}\_\infty+\norm{f'}\_\infty$
and let ${\mathcal I}$ be the closed ideal consisting of those $C^1$-functions which vanish at $x=0$ and whose 1st derivative vanishe... | 5 | https://mathoverflow.net/users/763 | 58778 | 36,545 |
https://mathoverflow.net/questions/58785 | 1 | Given probability distributions $(\mu\_1, \ldots, \mu\_n)$ on a nice state space $E$ is it always possible to find a random vector $(X\_1, \ldots, X\_n)$ such that $(X\_k, X\_{k+1})$ is an optimal coupling of $\mu\_k$ and $\mu\_{k+1}$ for any $1 \leq k \leq n-1$? For example, this is true for Gaussian distributions $\m... | https://mathoverflow.net/users/1590 | optimal coupling | The gluing Lemma.
Let $(X\_i , μ\_i)$, $i = 1, 2, 3$ be Polish probability spaces. If $(X\_1,X\_2 )$ is a coupling of $(μ\_1 , μ\_2 )$ and $(Y\_2 , Y\_3 )$ is a coupling of $(μ\_2,μ\_3 )$, then one can construct a triple of random variables $(Z\_1 , Z\_2 , Z\_3 )$
such that $(Z\_1 , Z\_2 )$ has the same law as $(X\_... | 2 | https://mathoverflow.net/users/12088 | 58786 | 36,551 |
https://mathoverflow.net/questions/58774 | 4 | I'm interested in whether Levin and Solomonoff's results on "universal semimeasures" can be extended to other settings. One case that especially interests me is finding "universal" strategies in the one-player game "guess the next bit, win 1 dollar if you're right, lose 1 dollar if you're wrong" played over infinite (a... | https://mathoverflow.net/users/13669 | Lower-semicomputable supermartingales with bounded increments | (Note: It is very possible I misunderstood the questions.)
By $X$ dominates $Y$ up to an additive constant, do you mean $X,Y$ are supermartingales with bounded increments, and $X(S)>Y(S)-C$ for some constant $C$ and all finite bit-strings $S$? If so, the answer to the first question is no.
Consider $M$ that "always... | 2 | https://mathoverflow.net/users/12978 | 58787 | 36,552 |
https://mathoverflow.net/questions/58732 | 5 | In cryptography one needs finite groups $G$ in which the discrete logarithm problem is infeasible. Often they use the multiplicative group $\mathbb{G}\_m(\mathbb{F}\_p)$ where $p$ is a prime number of bit length $500$, say.
Rubin and Silverberg suggested (cf. [1]) to use certain tori instead, if the goal is to minimi... | https://mathoverflow.net/users/8680 | Torus based cryptography | The discrete log problem in the multiplicative group of a finite field may be solved using the *index calculus*, not the number field sieve (although sieves are used to speed the process of checking numbers for smoothness during the index calculus algorithm). Anyway, the idea is as follows.
Using index calculus on a... | 11 | https://mathoverflow.net/users/11926 | 58792 | 36,556 |
https://mathoverflow.net/questions/58702 | 6 | Suppose $n$ is a large odd integer. Let $D\_1(n)$ be the number of divisors of $n$ of the form $4k+1$ and let $D\_3(n)$ be the number of divisors of the form $4k+3$. I would like to compute $(D\_1(n),D\_3(n))$.
As Joe Silverman points out, the number of representations of $n$ as a sum of two squares of integers is $... | https://mathoverflow.net/users/12922 | Number of divisors of an integer of form 4n+1 and 4n+3 | Not quite what you're asking, but an interesting theorem of Legendre's says that the number of ways of writing an integer $N$ as a sum of two squares is $4D\_1(N)-4D\_3(N)$, where $D\_1(N)$ is the number of positive divisors of $N$ that are congruent to 1 modulo 4 and $D\_3(N)$ is the number of positive divisors of $N$... | 12 | https://mathoverflow.net/users/11926 | 58795 | 36,557 |
https://mathoverflow.net/questions/58721 | 35 | There are many optimization problems in which the variables are symmetric in the objective and the constraints; i.e., you can swap any two variables, and the problem remains the same. Let's call such problems *symmetric optimization problems.* The optimal solution for a symmetric optimization problem - like many of the... | https://mathoverflow.net/users/9716 | When does symmetry in an optimization problem imply that all variables are equal at optimality? | The Monthly article "[Do symmetric problems have symmetric solutions?](http://www.maa.org/programs/maa-awards/writing-awards/do-symmetric-problems-have-symmetric-solutions)" by William Waterhouse discusses this issue. For global optimality one really needs strong global constraints on the objective function, such as co... | 45 | https://mathoverflow.net/users/4720 | 58798 | 36,559 |
https://mathoverflow.net/questions/58685 | 8 | Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?
A non connected counter example has been provided, so I'll ask for E,B and F to be connected (hopefully low dimensional) manifolds.
| https://mathoverflow.net/users/13700 | Trivial fiber bundle | Consider the pullback $\xi$ of $TS^2$ via the projection of $S^2\times\mathbb R$ onto the first factor. The bundle $\xi$ is a nontrivial $\mathbb R^2$-bundle over $S^2\times\mathbb R$ because its pullback under the inclusion $S^2\to S^2\times\mathbb R$ is $TS^2$, which is nontrivial. On the other hand, its total space ... | 13 | https://mathoverflow.net/users/1573 | 58802 | 36,560 |
https://mathoverflow.net/questions/58782 | 12 | This question is not about elements of $S\_n$ that consist of a single $n$-cycle, though naturally it's related.
Instead, consider permutations modulo the action of $(123\ldots n)$. That is, we want ABCD to be the same as BCDA and CDAB and DABC. (It's optional whether this also is the same as DCBA, but for now let's... | https://mathoverflow.net/users/12192 | Cyclic Permutations - but not what you think | They're called cyclic orders or cyclic orderings. Since they don't form a group, as you noticed, there's no representation theory. They play a role in the study of moduli spaces via ribbon graphs, for example. A ribbon graph is a graph with a cyclic ordering of the edges incident to every vertex. If you look for "cycli... | 7 | https://mathoverflow.net/users/4183 | 58805 | 36,561 |
https://mathoverflow.net/questions/58804 | 7 | The following is presumably not the greatest generality in which this question makes sense.
1. Given a ring $k$, graded-commutative if it helps, and a Hopf-algebra $A$ over $k$, there is a Yoneda product making $\textrm{Ext}\_A^\*(k, k)$ into a ring (since $k$ is graded, this actually is a bigraded object, but I susp... | https://mathoverflow.net/users/1191 | Ostensibly different products on Ext-groups | Show that each of those products distributes over the others, and use Hilton-Eckmann (over a field, or for $A$ projective; in general, I don't know...) This ends up being then an exercise in using naturality.
It can be done more concretely, too. For example, to show (1) and (2) are the same, one can show that if $E$ ... | 7 | https://mathoverflow.net/users/1409 | 58806 | 36,562 |
https://mathoverflow.net/questions/58738 | 2 | We are studying topology. There are a lot of definitions and theorems. I wonder if there somewhere knowledge base about topology and reasoning system exists. So I expect some tool that systematizes all things in topology and can answer me some simple questions about properties of topological object, give me list of the... | https://mathoverflow.net/users/13714 | Knowledge base about topology | I was working on a project like this about five years ago, but I abandoned it. I entered the tables from *Counterexamples in Topology* that list the topological spaces and their properties, and I created a list of logical dependencies among the properties. I also wrote Python code to search for spaces having specified ... | 2 | https://mathoverflow.net/users/4758 | 58809 | 36,565 |
https://mathoverflow.net/questions/58814 | 2 | Hey,
I need to numerically identify discontinuity points for a function given by a general expression (formula). I am able to evaluate the values at any point. I need it to be fast bu not accurate. The goal is to correctly render functions. With my naive algorithm, I get vertical lines on $x=0$ for $1/x$ and $sign(x)$... | https://mathoverflow.net/users/13737 | Numerically Identifying Discontinuity | It's going to be hard to find a "fast" way of doing this, but there is an [algorithm due to Jeff Tupper](http://dx.doi.org/10.1145/383259.383267) for reliably sketching discontinuous functions, which you should be able to adapt to your needs.
| 2 | https://mathoverflow.net/users/13738 | 58818 | 36,569 |
https://mathoverflow.net/questions/58825 | 15 | I asked this question on math.stackexchange yesterday, but nobody has helped so far, and only 44 people have seen it! So I hope people do not mind me asking it here...
Let $A$ be a finite abelian group, and let
$ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $
be an alternating, non-degenerate bilinear form on $A$... | https://mathoverflow.net/users/13741 | Non-degenerate alternating bilinear form on a finite abelian group | Actually, one can show the following stronger result:
>
> **Proposition.** Assume that a finite abelian group $A$ admits a non-degenarate, bilinear alternating form $\psi$. Then $A$ has a *lagrangian decomposition*, i.e. there exists a subgroup $G$, isotropic for $\psi$, such that $$A \cong G \times \widehat{G},$$ ... | 27 | https://mathoverflow.net/users/7460 | 58828 | 36,573 |
https://mathoverflow.net/questions/58776 | 3 | Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}\_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\lambda A$ for some $\lambda\in R$. Do we have $\mathrm{Tr}A=\lambda k$ ?
**Motivation:** This holds for $R$ a field, ... | https://mathoverflow.net/users/2530 | Pseudo-idempotent matrix generating a free module | $\DeclareMathOperator{\Tr}{Tr}$This is true. Here is an elementary proof: Let
$$ \phi\colon A\cdot R^n \to R^k \quad \text{and}\quad \psi\colon R^k\to A\cdot R^n $$
mutually inverse isomorphisms. Let $X$ be the matrix of the map $R^n\ni v\mapsto \phi(Av)\in R^k$, and $Y$ the matrix of $\psi$ as map from $R^k\to R^n$... | 2 | https://mathoverflow.net/users/10266 | 58832 | 36,576 |
https://mathoverflow.net/questions/58836 | 4 | In the great book by Harary and Palmer (Graphical Enumeration) one can find many interesting things about graph asymptotics.
For example it is stated that the number of all unlabeled graph is $\sim 2^{ n \choose 2}/n!$ and that almost all graphs are blocks. They also state that it's very likely that for all natural ... | https://mathoverflow.net/users/1737 | Connected graphs that are not 2 connected | See
<http://www.math.uwaterloo.ca/~nwormald/papers/2connected.pdf>
and references therein (in particular, to the Erdos/Renyi paper).
| 6 | https://mathoverflow.net/users/11142 | 58837 | 36,578 |
https://mathoverflow.net/questions/58813 | 3 | I am not specialist on Topological Group Theory, I apologize if this is a trivial question.
**Question.** If $G\_1=G\_2$ are amenable topological groups what additional hypothesis we have to consider on the group, in order to prove that $G\_1\times G\_2$ is amenable ?
Following Leinster, in this question [Why ar... | https://mathoverflow.net/users/2386 | If $G$ is amenable, when $G\times G$ is amenable ? | The wikipedia article is quite clear on the subject:
<http://en.wikipedia.org/wiki/Amenable_group>
| 3 | https://mathoverflow.net/users/11142 | 58841 | 36,580 |
https://mathoverflow.net/questions/58838 | 7 | I have been reading a number of papers by Jacques Tits (mostly written in the second half of 1980s) and in them he frequently refers to following publications of his:
1. Résumé de cours, Annuaire du collège de France, 81e année (1980-1981), 75-86.
2. Résumé de cours, Annuaire du collège de France, 82e année (1981-198... | https://mathoverflow.net/users/8811 | "Résumé de cours" by Jacques Tits | I've randomly acquired from Tits just a few of his resumes: 1979-80 (Bruhat-Tits work), 1980-81 (Kac-Moody algebras and groups), 1990-91 (Galois cohomology of semisimple groups over global fields). These are all quite short and not likely to add too much to related published papers, but if other means fail I could copy... | 10 | https://mathoverflow.net/users/4231 | 58846 | 36,582 |
https://mathoverflow.net/questions/58815 | 45 | A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever heard of the question before, and no one knew the answer.
I hope this is an original question, but seeing how natural it ... | https://mathoverflow.net/users/13700 | Polynomial roots and convexity | First, a counterexample to your conjecture. Let $\Pi = x^4+x^3+4x^2+4x = x(x+1)(x^2+4)$, so $P = 4x^3+3x^2+8x+4$. The critical values of $\omega$ are $1.06638, 3.89455 + 2.87687i, 3.89455 - 2.87687i$, and by inspection (using Mathematica) we see that for each of these values of $\omega$, $\mbox{Conv}(\Pi\_\omega)$ cont... | 18 | https://mathoverflow.net/users/2363 | 58851 | 36,585 |
https://mathoverflow.net/questions/58845 | 0 | Call two axioms equivalent if they imply the same set of theorems. I am interested in decidability of so defined equivalence. In this generality the problem is obviously undecidable since it can be used to decide Entscheidungsproblem. So I am interested in cases where Entscheidungsproblem is decidable, particularly in ... | https://mathoverflow.net/users/13745 | Equivalence of monadic axioms | The comment showed that decidability of axiom equivalence is implied by decidability of pure logic. (I.e. to decide if $\Theta$ and $\Theta'$ are equivalent, it is equivalent to decide if $\vdash \Theta \leftrightarrow \Theta'$. Conversely, if one can decide axiom equivalence, then one can decide pure logic. Namely, $\... | 1 | https://mathoverflow.net/users/9896 | 58853 | 36,586 |
https://mathoverflow.net/questions/58856 | 16 | A fundamental result in Diophantine approximation, which was largely responsible for Klaus Roth being awarded the Fields Medal in 1958, is the following simple-to-state result:
If $\alpha$ is a real algebraic number and $\epsilon > 0$, then there exists only finitely many rational numbers $p/q$ with $q > 0$ and $(p,q... | https://mathoverflow.net/users/10898 | Advances and difficulties in effective version of Thue-Roth-Siegel Theorem | The non-effectivity, as far as I understand, is already present in Thue's Theorem, thus to understand it, one can look a the proof of the latter. The issue is roughly that, to show that there are not many "close rational approximations" $p/q$, one starts with the assumption that there exists *one* very close one $p\_0/... | 21 | https://mathoverflow.net/users/20038 | 58860 | 36,592 |
https://mathoverflow.net/questions/58812 | 27 | It's an obvious consequence of the pigeonhole principle that any injective function over finite sets is bijective. But there are some similar results in different areas of mathematics that apply to less-finite settings.
In algebraic geometry, the [Ax–Grothendieck theorem](http://en.wikipedia.org/wiki/Ax%E2%80%93Groth... | https://mathoverflow.net/users/440 | Ax–Grothendieck and the Garden of Eden | See the recent paper ["On algebraic cellular automata"](http://www.arxiv.com/abs/1011.4759), for a proof of how to derive the Garden of Eden theorem from the Ax-Grothendieck theorem. This is indeed in the spirit of Gromov's paper that Mohan mentioned in the comments. This paper is were he introduced Sofic groups. The s... | 15 | https://mathoverflow.net/users/2384 | 58865 | 36,595 |
https://mathoverflow.net/questions/58870 | 23 | I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I am fo... | https://mathoverflow.net/users/5337 | What should be taught in a 1st course on smooth manifolds? | I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sadly neglected in introductory courses and books on manifolds. It is completely elementary: witness [these lecture notes]... | 26 | https://mathoverflow.net/users/450 | 58884 | 36,603 |
https://mathoverflow.net/questions/57751 | 4 | Is there a known correlation structure among the maximums of a Brownian motion on disjoint intervals ?
Let $(W\_t)\_{t\geq 0}$ be a one-dimensional standard Brownian motion,
and take the partition $0=t\_0< t\_2 <...< t\_i, t\_j <...t\_N=T$ and define the maximum of the Brownian motion on the interval $[t\_i,t\_{i+1... | https://mathoverflow.net/users/13486 | Correlation structure among the maximums of a Brownian motion | You ask for the correlation function, defined for every $0\le s\le t\le u\le v$ by the formula
$$
C(s,t;u,v)=E(Y\_{s,t}Y\_{u,v})-E(Y\_{s,t})E(Y\_{u,v}).
$$
One first computes $E(Y\_{s,t})$. For every $t\ge0$, one introduces
$$
M\_t=\max\{B\_s;0\le s\le t\},
$$
where $(B\_t)$ is another standard Brownian motion, indep... | 7 | https://mathoverflow.net/users/4661 | 58888 | 36,605 |
https://mathoverflow.net/questions/58844 | 3 | Hello,
let's define the notion of order of primality in such a way:
$n$ is a prime of order $k$ if and only if $k$ is the smallest non negative integer such that $\pi^{(k)}(n)$ is not a prime number, where $\pi^{(0)}(n)=n$ and for all $m$, $\pi^{(m+1)}(n)=\pi(\pi^{(m)}(n))$.
Now let's write $\pi\_{k}(x)$ for the numb... | https://mathoverflow.net/users/13625 | order of primality of an integer | I think your argument may be generalized to any order. If $m \in \mathbb{N}$, then $\pi^{(m)}(x)$ is the number of primes of order $\ge k$ not exceeding $x$. That is
$$\pi^{(m)}(x) = \sum\_{k \ge m} \pi\_k(x)$$
And we get
$$\pi\_m(x) = \pi^{(m)}(x) - \pi^{(m+1)}(x)$$
Thus for $n \in \mathbb{N}$, we get
$$\pi\... | 1 | https://mathoverflow.net/users/13263 | 58889 | 36,606 |
https://mathoverflow.net/questions/58874 | 5 | I re-read Jechs chapter about forcing, and got a question. There he characterizes a (what he calls) modern way to make the forcing argument legitimate which (I think) goes like this:
It is pointed out there that, in order to establish the consistency of a statement $\varphi$ relative to ZFC, it is sufficient to exhi... | https://mathoverflow.net/users/8996 | Generic filter over $V$ | The existence of a generic $G$ over $V$ is indeed impossible in $V$ (for nontrivial forcing notions), but it has truth value 1 in appropriate Boolean-valued models. In more detail: If $P$ is a partially ordered set (to be used as a notion of forcing) and $B$ is the complete Boolean algebra of regular open subsets of $P... | 12 | https://mathoverflow.net/users/6794 | 58896 | 36,612 |
https://mathoverflow.net/questions/58917 | 5 | Suppose we have a countable set S with a total order. Can we give an injection from S to the set of finite binary sequences that end in all zeros that preserves the ordering? The order on binary sequences is the dictionary ordering (e.g. 001001 <= 01).
For a finite set this is easy: arrange the set in order and assig... | https://mathoverflow.net/users/6210 | Injections to binary sequences that preserve order | It is easy to prove that for any countable linearly ordered set there is an order preserving injection to the rationals. This can be proven by enumerating the base set and then specifying the values of the mapping by induction.
Since you have a solution for $\mathbb Q$, for other sets just compose the order preservi... | 6 | https://mathoverflow.net/users/2384 | 58919 | 36,624 |
https://mathoverflow.net/questions/58912 | 3 | For $n\in \mathbb{N}$ numbers $I\_{n}=(1,2,3..n)$ and prime $p$, we define operation $(1,2,3..n)$ to $A=(a\_{1},a\_{2}...a\_{p-1})$ as follows:
We arrange the $n$ numbers in a circle, then we eliminate the first number, the $p$th number, the 2$p$th number, etc, until there is only $p-1$ numbers left and the process t... | https://mathoverflow.net/users/5175 | Is this set equidistributed? | Not quite an answer to the question but see <http://en.wikipedia.org/wiki/Josephus_problem>
There is a huge literature on the Josephus problem -- the wiki article is a good start. See also:
<http://doc.utwente.nl/67513/1/pospp.pdf>
and the very cool:
<ftp://ftp.cis.upenn.edu/pub/wilf/josephus.ps>
| 3 | https://mathoverflow.net/users/11142 | 58923 | 36,627 |
https://mathoverflow.net/questions/58555 | 27 | Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ 1&2&2\end{pmatrix}$, and $C = \frac{1}{5}\begin{pmatrix}2&1&2\\\ 1&2&2\\\ 0&0&5\end{pmatrix}$.
The group $G\subset GL\_3(\mathbb{C})$ they generate is free of rank 3. However, o... | https://mathoverflow.net/users/12301 | Understanding "infinite" relations in groups | This answer is partly an expansion of the comments so far. What you are looking for is a type of decoration on groups that (1) lets you evaluate infinite products, and (2) also creates extra points that are the value of some or all of those products. There is one fairly inevitable answer to (1): You should consider top... | 17 | https://mathoverflow.net/users/1450 | 58930 | 36,631 |
https://mathoverflow.net/questions/58669 | 4 | Let $X$ be a projective variety of dimension $d$ over $k=\bar{k},$ with $L$ an ample line bundle on $X$ and $\eta=c\_1(L).$ Hard Lefschetz gives an isomorphism (see BBD)
$$
\eta^i:IH^{d-i}(X)\to IH^{d+i}(X)
$$
with Tate twist ignored, which, together with the intersection pairing between $IH^{d-i}$ and $IH^{d+i},$ give... | https://mathoverflow.net/users/370 | intersection pairing on intersection cohomology | You are right that this symmetry follows from a similar formula on the complex
level. To begin with $\eta^i$ is induced from multiplication by $c\_1(\mathcal
L)^i$ in $H^\ast(X)$ and the $H^\ast(X)$-module structure on the intersection
cohomology. Hence your result will follow from the fact that the Poincaré
pairing is... | 5 | https://mathoverflow.net/users/4008 | 58933 | 36,633 |
https://mathoverflow.net/questions/58883 | 10 | Let $K=\lim(K\_{i})$ be an ultrafield (over a non-principal ultrafilter), and let $K\hookrightarrow K'$ be a field extension of $K$.
When the field $K'$ is finite over $K$ it is also an ultrafield by Łoś's theorem. What can be said when the trascendence degree of $K'$ over $K$ is infinite?
| https://mathoverflow.net/users/12940 | Is every field extension of an ultrafield an ultrafield? | [EDIT on March 26, 2019: There are issues with this argument, found by @YCor who also gives a correct and more general answer.]
Let me show that if $k$ is algebraically closed, and $X=(x^{(\alpha)})$ any nonempty family of indeterminates, then $k(X)$ is not an ultrafield (which provides a lot of counterexamples since... | 9 | https://mathoverflow.net/users/7666 | 58934 | 36,634 |
https://mathoverflow.net/questions/58939 | 2 | I have been looking for references on sheaves that take value in the category of Boolean rings (e.g. about cohomology, etc). Would someone be able to give me some?
Or are they interesting at all? Thanks!
| https://mathoverflow.net/users/7150 | References for Sheaves of Boolean Rings | This is not exactly what you're asking for, but there are some results about cohomology groups of sheaves on Stone spaces. But it is probably related to your question since sheaves with values in Boolean rings are only interesting when the space is disconnected.
The paper "Sheaf cohomology of locally compact totally ... | 2 | https://mathoverflow.net/users/2841 | 58940 | 36,637 |
https://mathoverflow.net/questions/58957 | 3 | Hello, all!
I consider Hadamard product $A \circ B$ of matrices $A$, $B$ over finite field. I know $\det{A}$ and $\det{B}$ and want to know about $\det{(A \circ B)}$. Wikipedia and Google let me know properties about determinant for Hadamard product of positive-semidefinite matrices: $det{(A \circ B)} \ge \det{A} \cd... | https://mathoverflow.net/users/nan | Hadamard product of matrices over finite field | Given any $a$ and $b$, let $A=\pmatrix{a&0\cr0&1}$, $B=\pmatrix{0&-b\cr1&0}$, then $\det A=a$, $\det B=b$, and the determinant of the Hadamard product is zero.
| 3 | https://mathoverflow.net/users/3684 | 58959 | 36,647 |
https://mathoverflow.net/questions/58960 | 0 | Suppose $\nu$ is a regular signed or complex Borel measure on $\mathbb R^n$, *m* is the Lebesgue measure on the class of Borel sets $\mathcal B\_{\mathbb R^n}$ and the Lebesgue-Radon-Nikodym decomposition of $\nu$ is $d\nu=d\lambda+fdm$ where *f* is an extended *m*-integrable function when $\nu$ is a signed measure or ... | https://mathoverflow.net/users/5072 | A question about regular signed or complex Borel measure under LRN decomposition | If $\nu\bot\lambda$, then $|\nu+\lambda|=|\nu|+|\lambda|$
| 1 | https://mathoverflow.net/users/13776 | 58962 | 36,648 |
https://mathoverflow.net/questions/58964 | 6 | I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers).
1. Is a universal homeomorphism of connected regular (excellent finite dimensional) schemes an isomorphism if these schemes are not positive char... | https://mathoverflow.net/users/2191 | More on universal homeomorphisms | 1. Yes. Let $f\colon X \to Y$ be a universal homeomorphism of locally noetherian schemes. Assume that $X$ and $Y$ are integral and $Y$ is normal, and the function field $k(Y)$ has characteristic 0. Then $k(X) = k(Y)$, and $f$ is an isomorphism by Zariski's main theorem.
2. Under these conditions $f$ is a universal home... | 8 | https://mathoverflow.net/users/4790 | 58965 | 36,649 |
https://mathoverflow.net/questions/58850 | 2 | I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. Unfortunately i don't have a deeper insight into this subject, so i don't know how things behave in more complicated situations, like i ran into:
Given a regular local ring $A$ of dimension $\leq 2$ and an ... | https://mathoverflow.net/users/3233 | Torsion in tensor products over noncommutative rings | Instead of updating my previous answer, I've decided to add a new answer in order to keep it short(ish).
In the comments following his original question, TonyS added the extra assumption that $R$ is finitely generated over $A$. This is a strong condition, since it makes $R$ very close to being commutative. Moreover $... | 2 | https://mathoverflow.net/users/6827 | 58969 | 36,652 |
https://mathoverflow.net/questions/58926 | 5 | Is it well known what happens if one blows-up $\mathbb{P}^2$ at points in non-general position (ie. 3 points on a line, 6 on a conic etc)? Are these objects isomorphic to something nice?
| https://mathoverflow.net/users/13766 | Blow-ups at points in non-general position | I'll just add to Francesco's answer by saying that general position of the points on the plane is equivalent to ampleness of the anticanonical sheaf $\omega\_X^{\otimes -1}$.
The key observation is that on a del Pezzo surface, an irreducible negative curve ($C^2 < 0$) must be an exceptional curve (i.e. $C^2 = C\cdot ... | 2 | https://mathoverflow.net/users/392 | 58973 | 36,655 |
https://mathoverflow.net/questions/58966 | 15 | It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.
The proof I know goes as follows:
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension for a suitable homogeneous differential equation for which $f$ is a solution.
Then one may calculate $G(E/F)$ and show... | https://mathoverflow.net/users/3759 | Solvability in differential Galois theory | The analogue to "solvable by radicals" in differential Galois theory is "solvable by quadratures". The theorem says that a PV-extension is Liouvillian (adjoining primitives and exponentials) iff the connected component of the differential Galois group is solvable. See ["A first look at differential algebra"](http://www... | 11 | https://mathoverflow.net/users/2384 | 58974 | 36,656 |
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