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https://mathoverflow.net/questions/53445 | 3 | I was reading [these slides](http://www.cl.cam.ac.uk/~jrh13/slides/cmu-19mar07/slides.pdf) by John Harrison, and was struck by the comment at the end about the universal fragment of real-closed fields needing nothing more than the axioms for an (ordered) integral domain. Since an obvious integral domain is the integers... | https://mathoverflow.net/users/202 | Real-closed fields minus existentials for Presburger-like power and multiplication? | The universal theory of arithmetic, in the language with $+$ and $\cdot$, is not decidable, because this is exactly sufficient to ask whether a given diophantine equation has no solutions in the integers, and this is not decidable by the [MRDP theorem](http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem), which solv... | 4 | https://mathoverflow.net/users/1946 | 53447 | 33,430 |
https://mathoverflow.net/questions/53440 | 6 | Hi,
could anybody recommend a CAS suited to DG/GR applications such as computation of connection coefficients or generating (and possibly solving) PDEs for, for example, an unknown metric of given curvature. Oh, and compatible Linux (I'm using Maple through wine but am having myriad problems. Also tried Maxima but I ... | https://mathoverflow.net/users/4890 | Differential Geometry/General Relativity Computer Algebra | Mathematica has had GR stuff for decades (here is a random link:
<http://wps.aw.com/aw_hartle_gravity_1/0,6533,512496-,00.html>
but google search for Mathematica "general relativity" returns lots.
I don't understand your comment about Maple -- it certainly has a linux version.
| 4 | https://mathoverflow.net/users/11142 | 53451 | 33,434 |
https://mathoverflow.net/questions/53470 | 9 | Is there a standard reference for the theory (if it exists) of $\mathcal{V}$-enriched locally presentable categories? Here $\mathcal{V}$ is a cosmos. Does anything unexpected happens here in contrast to the case $\mathcal{V}=\text{Set}$ treated in "Locally Presentable And Accessible Categories" by Adamek & Rosicky? In ... | https://mathoverflow.net/users/2841 | Enriched locally presentable categories | The standard reference is Max Kelly’s 1982 paper [“Structures defined by finite limits in the enriched setting”](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1982__23_1/CTGDC_1982__23_1_3_0/CTGDC_1982__23_1_3_0.pdf).
Perhaps the most unexpected thing is how well the theory works!
| 17 | https://mathoverflow.net/users/10862 | 53474 | 33,446 |
https://mathoverflow.net/questions/53465 | 0 | Suppose M is a compact Lie Group, is there a Schauder basis for L^1(M)?
| https://mathoverflow.net/users/12444 | About Schauder Basis | Every separable $L\_1$ space is isomorphic to $\ell\_1$ or $L\_1(0,1)$ and thus has a Schauder basis. Look at, for example, Classical Banach spaces by Lindenstrauss and Tzafriri. Other books probably have it, too; see Albiac-Kalton Topics in Banach space theory or Wojtaszczyk's Banach spaces for analysts.
| 11 | https://mathoverflow.net/users/2554 | 53477 | 33,448 |
https://mathoverflow.net/questions/53469 | 3 | The number of increasing paths from (0,0) to (n,m) with only vertical (north) and horizontal (east) moves can be easily proved to be $\binom{n+m}{n}$. When adding the possibility of making diagonal (north-east) moves, I get that the total number of possible paths is $F(n,m)=\sum\_{p=\max(n,m)}^{n+m}\binom{p}{n+m-p, p-m... | https://mathoverflow.net/users/9804 | concise formula for number of paths from (0,0) to (n,m) with horizontal, vertical and diagonal moves? | These are [Delannoy numbers](http://mathworld.wolfram.com/DelannoyNumber.html) [A008288](http://oeis.org/A008288).
One of the ways they arise is as the count of domino tilings of a modified Aztec diamond. Then the Lindstrom-Gessel-Viennot theorem says that the number of domino tilings of an [Aztec diamond](http://en... | 6 | https://mathoverflow.net/users/2954 | 53478 | 33,449 |
https://mathoverflow.net/questions/53055 | 1 | Let $d\_c, \delta\_c$ be operators with domains $D(d\_c) = D(\delta\_c) = C\_{c}^\infty(\wedge T^\ast M)$. We let $d\_c$ be the usual exterior derivative on compactly supported smooth forms, ie., $d\_c\omega = dx^k \wedge \nabla\_k \omega$ and $\delta\_c = dx^k \llcorner \nabla\_k \omega$.
We define $d:D(d) \subset L... | https://mathoverflow.net/users/11976 | Commutativity of pullbacks and the exterior derivative as an unbounded operator on $L^2$ | Fix $u,v \in C\_c^\infty$. Then, let $\psi = \phi^{-1}$. Then, let $(\psi^\ast)^\ast$ be the adjoint of $\psi^\ast$. So,
$$\langle u, (\psi^\ast)^\ast\delta\_c v \rangle
= \langle \psi^\ast u, \delta\_c v \rangle
= \langle d\psi^\ast u, v \rangle
= \langle \psi^\ast d u, v \rangle
= \langle d \psi^\ast u , v \rangle
= ... | 0 | https://mathoverflow.net/users/11976 | 53481 | 33,452 |
https://mathoverflow.net/questions/53454 | 11 | Consider some probability distribution $D$ over non-negative reals with finite expectation $\mu$. Now for any positive $T$ consider sums of $T$ iid random variables drawn from $D$. A single sum of this sort would be $S(T) = \sum\_{i = 1}^T x\_i$ where each $x\_i$ is a iid random sample from $D$.
We will consider $n$ ... | https://mathoverflow.net/users/5873 | Maximum of a set of sums of iid random variables | It is always true. Split $x\_i=y\_i+z\_i$ where $y\_i$ are bounded and $Ez\_i\le \frac \mu{10n}$. You have no problems with $y\_i$ because if they were alone,$ES\_j$ would be concentrated in a very strong sense around $\nu T$ for large $T$ where $\nu=Ey\_i\le\mu$ (see Didier's argument for details or recall the Bernste... | 13 | https://mathoverflow.net/users/1131 | 53483 | 33,454 |
https://mathoverflow.net/questions/53475 | 4 | I wonder whether it is possible to have a smooth action of the $ax+b$ Lie group with compactly supported fundamental vector fields on $\mathbb{R}^2$ in such a way that it is non-trivial at least at one point, i.e.linearily independent fundamtental vector fields or so.
The abelian case is possible (also in higher dime... | https://mathoverflow.net/users/12482 | Action of $ax+b$ with compact support | No, it's not possible, because of the generalized Reeb Stability Theorem,
*A generalization of the Reeb Stability Theorem*, William P. Thurston, Topology, V13,
pp 347--352, 1974. The theorem basically says that for any group of $C^1$-smooth diffeomorphisms
of a manifold that has a fixed point where every element has fi... | 9 | https://mathoverflow.net/users/9062 | 53484 | 33,455 |
https://mathoverflow.net/questions/53266 | 7 | Highly grateful for your help/steers on the following question (at the end):
Take the infinite product:
$$\displaystyle T(s) = \prod \_{n=2}^{\infty } \left( \dfrac{{n}^{s}} {{n}^{s}-1}\right)$$
for $\Re(s) > 1$ it is equal to:
$$\displaystyle \prod \_{primes}^{\infty } \left( \dfrac{{p}^{s}} {{p}^{s}-1}\right)... | https://mathoverflow.net/users/12489 | Values where infinite products of primes and composites are equal | I think that T is meromorphic on $\mathbb{C}$ just like $\zeta$, with a single pole at $s=0$. The ratio should be fine everywhere except at $s=1$, the negative integers, and the critical strip (or line, on the RH).
| 2 | https://mathoverflow.net/users/6043 | 53494 | 33,463 |
https://mathoverflow.net/questions/53496 | 3 | Hi,
if we could write a classification about the known regularity which is the known class of schemes that are immediately less good than normal schemes? And which properties have they?
thank you
| https://mathoverflow.net/users/12543 | less than normal | "Immediately less good than normal schemes" probably does not make much sense, since "goodness" depends on what you want to do.
For instance, often one likes normal varieties since it is possible to define on them a canonical (Weil) divisor $K$. This does not depend really on normality, but only on the fact that any ... | 5 | https://mathoverflow.net/users/7460 | 53499 | 33,466 |
https://mathoverflow.net/questions/53501 | 5 | Grothendieck Existence, which I imagine is the less well known result among the two, states the following:
Let $A$ be a noetherian ring that is complete w.r.t. a proper ideal $I$. Let $V$ be a proper $A$-scheme. Let $W$ be the inverse image of the locus of $I$ (as a subscheme of $V$). Let $\mathfrak{V}=(W,\mathcal{O}\_... | https://mathoverflow.net/users/5309 | How does Schlessinger's criterion sit with Grothendieck Existence (aka GFGA)? | Schlessinger's criterion is a criterion for the pro-representability of a functor. This is the same thing is getting something like an object on the formal scheme $\mathfrak V$. Grothendieck's result tells us that this object comes from an actual object on $V$. Usually this is the step after pro-representability to get... | 10 | https://mathoverflow.net/users/4008 | 53508 | 33,472 |
https://mathoverflow.net/questions/53378 | 14 | I know of two conjectural descriptions of the motivic $t$-structure for Voevodsky's motives.
The first one is based on the conjecture that Weil cohomology theories should yield exact and conservative functors on the category of mixed motives. In the paper
Hanamura M. Mixed motives and algebraic cycles, III// Math... | https://mathoverflow.net/users/2191 | Are two conjectural descriptions of the motivic t-structure (via cohomology and via affine varieties) known to be equivalent? | Okay, I'm not familiar with Beilinson's paper, but here's my take on this. First let's recall the two definitions. I will denote the triangulated category of motives over a field $k$ by $DM(k)$ (for any of the equivalent definitions that are available); I am taking $\mathbb{Q}$-coefficients and looking only at compact ... | 7 | https://mathoverflow.net/users/12336 | 53509 | 33,473 |
https://mathoverflow.net/questions/53503 | 5 | Update #3:
Over on *TCS StackExchange*, I have rated as "accepted" [an ingenious construction by Luca Trevisan](https://cstheory.stackexchange.com/questions/4704/do-runtimes-for-p-require-exp-resources-to-upper-bound-are-concrete-examples-k/4716#4716), which answers a two-part question (as reframed by Tsuyoshi Ito) t... | https://mathoverflow.net/users/11394 | Does BQP^P = BQP ? ... and what proof machinery is available? | Yes, P is contained in BQP (Benioff, 1982; <http://prl.aps.org/abstract/PRL/v48/i23/p1581_1> ) and $BQP^{BQP}=BQP$, for pretty much the same reason $BPP^{BPP}=BPP$. This second point first appeared (that I know of) as Cor 4.15 of BBBV'97: <http://www.cs.berkeley.edu/~vazirani/pubs/bbbv.ps> .
| 7 | https://mathoverflow.net/users/7718 | 53513 | 33,475 |
https://mathoverflow.net/questions/53514 | 2 | What are the irreducible solutions of Seiberg-Witten equation on S^2\times S^1?
Thanks.
| https://mathoverflow.net/users/12445 | Seiberg-Witten equation on S^2\times S^1 | The equations depends on a Riemannian metric and on a perturbation term (a closed 2-form). The usual metric on $S^2\times S^1$ has positive scalar curvature, which implies that there are no irreducible solutions when the 2-form vanishes.
This is a consequence of the Weitzenboeck formula for the Dirac operator, as wa... | 5 | https://mathoverflow.net/users/2356 | 53519 | 33,478 |
https://mathoverflow.net/questions/53498 | 26 | There is a famous circular argument for the Prime Number Theorem (PNT). It turns
out that there exists an infinite sequence of elementary-to-prove Chebyshev-type estimates
that taken together imply PNT. Unfortunately, the *collective* existence of all these proofs seems to require the PNT, so one must work hard a la Se... | https://mathoverflow.net/users/10909 | Nontrivial circular arguments? | Perhaps an example of the kind of circularity you mention
arises with the self-reference phenomenon that arises in
connection with the [incompleteness
theorems](http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)
and related applications. Specifically, Gödel proved
the fixed-point lemma that for any as... | 34 | https://mathoverflow.net/users/1946 | 53522 | 33,480 |
https://mathoverflow.net/questions/53530 | 8 | Hello!
Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general unramified extensions, is there a bound (depending on $K$) on the degree of an unramified extension over $K$? If so, does the co... | https://mathoverflow.net/users/3680 | Maximal (non-abelian) extensions of number fields unramified everywhere | No- even the process of iteratively taking the Hilbert class field, the Hilbert class field of the Hilbert class field, etc, need not terminate. See
<https://en.wikipedia.org/wiki/Golod-Shafarevich_theorem>
| 11 | https://mathoverflow.net/users/1018 | 53531 | 33,484 |
https://mathoverflow.net/questions/53527 | 33 | Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a
field (perfect, say). Assume that $R \otimes\_F \overline F$ is a polynomial ring over the
algebraic closure $\overline F$.
Does it follow that $R$ was already a polynomial ring over $F$?
I doubt it, but haven't had any luck constructing a counterexam... | https://mathoverflow.net/users/391 | If a field extension gives affine space, was it already affine space? | In his [Bourbaki talk of 1994](http://www.numdam.org/article/SB_1994-1995__37__295_0.pdf) Kraft tells us that the complex affine plane does not have non-trivial forms and that the corresponding question in higher dimensions is open. 1994 is an age ago, though!
(He also shows that the automorphism groups in high dimen... | 18 | https://mathoverflow.net/users/1409 | 53538 | 33,487 |
https://mathoverflow.net/questions/53537 | 2 | I need one to test a theory. There are probably many, but I can't seem to think of a single one. My guess is examples are pretty big.
Is there a systematic way to find such examples? Are there databases one can go through to find these things?
| https://mathoverflow.net/users/5309 | What is an example of a finite centerless group with at least 3 generators? | The group $S\_3\wr ({\mathbb Z}\_2 \times {\mathbb Z}\_2 \times {\mathbb Z}\_2)$ where $S\_3$ is the symmetric group with 6 elements, ${\mathbb Z}\_2$ is the group with 2 elements, $\wr$ is the wreath product.
The fact that it does not have a center is proved by inspection. The fact that it needs at least 3 generato... | 12 | https://mathoverflow.net/users/nan | 53540 | 33,489 |
https://mathoverflow.net/questions/53515 | 17 | Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?
I know the general problem of a set of polygons is undecidable, but I haven't found any information on the single tile case.
| https://mathoverflow.net/users/12249 | Decidability of tiling R^2 | If I remember right any Turing machine can be translated into a set of Wang tiles, so that the tiling problem for this set of tiles is equivalent to the decidability of the turing machine.
Assuming the non-existence of an aperiodic set of tiles, Wang provided the Decidability of any tiling problem in R^2. Since this ... | 16 | https://mathoverflow.net/users/11552 | 53547 | 33,491 |
https://mathoverflow.net/questions/53427 | 1 | Hi. I have read that stably free modules not finitely generated are free; this is proved in
M.R. Gabel, *stably free projectives over commutative rings*, Thesis, Brandeis Univ., Waltham, MA 1972.
But I can't find a proof of that, can anyone of you help me?
| https://mathoverflow.net/users/12532 | Stably free module not finitely generated is free | I think that the proof contained in <http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/stablyfree.pdf> (link given by darij grinberg) it's the best proof. Thanks to all
| 1 | https://mathoverflow.net/users/12532 | 53552 | 33,493 |
https://mathoverflow.net/questions/53431 | 54 | This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those given in [this handout](http://www.imomath.com/tekstkut/funeqn_mr.pdf). While I can see the pedagogical value in doing a few... | https://mathoverflow.net/users/290 | Does any research mathematics involve solving functional equations? | In additive combinatorics, one often seeks to count patterns such as an arithmetic progression $a, a+r, \ldots, a+(k-1)r$. When doing so, one is naturally led to expressions such as
$$ {\bf E}\_{a,r \in G} f\_0(a) f\_1(a+r) \ldots f\_{k-1}(a+(k-1)r)$$
for some finite abelian group $G$ and some complex-valued functions ... | 43 | https://mathoverflow.net/users/766 | 53558 | 33,497 |
https://mathoverflow.net/questions/53419 | 7 | The following fairly specific question comes up in a bordism computation I'm trying to do:
Are there compact $\mathbb Z$-oriented $4k+2$ dimensional manifolds with boundary $M$ such that $im(H\_{2k+1}(M; \mathbb Z/2)\to H\_{2k+1}(M, \partial M; \mathbb Z/2))$ has odd dimension as a $\mathbb Z/2$ vector space?
Clear... | https://mathoverflow.net/users/6646 | Are there oriented $4k+2$ manifolds such that $im(H_{2k+1}(M; Z/2)\to H_{2k+1}(M, \partial M; Z/2))$ has odd dimension? | I claim it is not possible. The image is the rank of $H\_{2k+1}(M;\mathbb Z\_2)/rad$, where
$rad$ is the radical of the intersection form on $H\_{2k+1}(M;\mathbb Z\_2)$.
The intersection form on $H\_{2k+1}(M;\mathbb Z\_2)/rad$ is hyperbolic, i.e. has a "symplectic" basis, therefore this vector space has even dimensi... | 8 | https://mathoverflow.net/users/1090 | 53559 | 33,498 |
https://mathoverflow.net/questions/53587 | 2 | Let $E$ be a vector bundle of rank $3$ on a smooth, projective surface and let $\varphi:E\to E$ be a nilpotent endomorphism. If $E$ is indecomposable, is it true that $\varphi$ has rank $2$? My claim is suggested by the fact that, in the case of a vector space $V$ of dimension $3$, if $\varphi:V\to V$ is an automorphis... | https://mathoverflow.net/users/33841 | Nilpotent endomorphisms of indecomposable vector bundles on a surface. | The answer is no, in fact you can already construct a counter-example on an elliptic curve $E$:
A non-zero element in $\mathrm{Ext}^1(O\_E, O\_E)$ produces th unique indecomposable 2-dimensional vector bundle $F$ (the Atiyah bundle) fitting into the short exact sequence
$O\_E \into F \onto O\_E$. A short exact sequen... | 2 | https://mathoverflow.net/users/7437 | 53590 | 33,511 |
https://mathoverflow.net/questions/53583 | 9 | All rings are Noetherian and commutative, modules are finitely generated.
It is a theorem of Serre that over a regular ring $R$, every module has a finite projective resolution.
More generally, if $R$ is regular in codimension n, what can we say about projective resolution of modules over $R$? For example, is it tr... | https://mathoverflow.net/users/8932 | Projective resolution of modules over rings which are regular in codimension n | If $R$ is normal (so regular in codimension $1$), excellent and local and all height $1$ ideals $I$ have finite projective dimension, then $R$ is factorial. So there are many counter-examples. (I don't have a reference to hand, but the argument is Serre's proof that regular implies factorial. Say $X= Spec\ R$ and $j:U\... | 7 | https://mathoverflow.net/users/8726 | 53598 | 33,514 |
https://mathoverflow.net/questions/53580 | 4 | Let $Y$ be an algebraic variety over a field $K$ whose coordinate ring is given by $K[Y]=K[X\_1,...,X\_n]/(F)$, where $F=\prod\_{i=1}^{n} X\_{i}^{a\_i}-1$ is an element in the polynomial ring $K[X\_1,...,X\_n]$, with each $a\_i$ an integer.
Now if we assume that the greatest common divisor $gcd(a\_1,...,a\_n)=1$, then... | https://mathoverflow.net/users/12566 | The irreducibility of an algebraic variety | There is nothing wrong with Qing's proof of irreducibility but the following is perhaps more conceptual (though less elementary). We immediately reduce to the case when all the $a\_i$ are different from zero. Then in the quotient ring, the images of the variables are invertible so we may replace the polynomial ring wit... | 3 | https://mathoverflow.net/users/4008 | 53602 | 33,517 |
https://mathoverflow.net/questions/53603 | 4 | Has $F(n)=\prod\_{p\ {\rm prime}, p-1|n}p$ been studied?
This function interests me because for each prime $p$, the long-term average number of factors of $p$ in $n$ equals the long-term average number of factors of $p$ in $F(n)$.
Any non-trivial result would interest me, particularly the distribution of values of ... | https://mathoverflow.net/users/10909 | Has $F(n)=\prod_{p\ {\rm prime}, p-1|n}p$ been studied? | I think $F(n)$ is (more-or-less) the denominator of the Bernoulli number $B\_n$ by the Von Staudt–Clausen theorem, see <http://en.wikipedia.org/wiki/Von_Staudt>–Clausen\_theorem which refers you to <http://oeis.org/A002445>
| 5 | https://mathoverflow.net/users/3684 | 53606 | 33,520 |
https://mathoverflow.net/questions/53557 | 12 | Hello,
this question may be simple but I couldn't find a reference.
Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C\_b^{\omega}(\Omega,F)$ be the vector space of bounded real analytic functions from $\Omega$ to $F$. Now I would like to know if there is a natural way to define a ... | https://mathoverflow.net/users/12558 | metric on the space of real analytic functions | The problem is nontrivial already in the finite dimensional case $E= \mathbb R^d$, $F=\mathbb R$. The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^d$ does not have any obvious or natural metric which would make it a Fréchet space.
The good news is... | 15 | https://mathoverflow.net/users/5371 | 53623 | 33,527 |
https://mathoverflow.net/questions/53597 | 1 | Let $Y$ be a connected CW-complex and $X\subset Y$ a connected CW-subcomplex. Suppose that each cell of $X$ is the boundary of a cell of $Y$. Is this enough to conclude that $X$ is contractible in $Y$ (the inclusion map is homotopic to a constant map)? If the answer is no, then which condition could be enough to get th... | https://mathoverflow.net/users/12570 | Contractibility of CW subcomplexes | If you mean that every cell in $X$ is the image of the boundary of a cell in $Y$ then the answer is no -- consider for example the standard inclusion $S^1 = RP^1 \hookrightarrow RP^2$. The boundary of the $2$-cell is $RP^1$ (but it runs around twice, so $\pi\_1 (RP^2) = \mathbb{Z}/2$ and the $1$-cell represents the gen... | 4 | https://mathoverflow.net/users/4183 | 53626 | 33,528 |
https://mathoverflow.net/questions/53511 | 12 | Let $(A,\mathfrak{m}\_A)$ be a local Artinian $k$-algebra with residue field $k$. Then the scheme $\mathrm{Spec}(A)$ can be loosely seen as a "fat point", or an "infinitesimal neighbourhood" of a point $\mathrm{Spec}(k)$. If the maximal ideal satisfies the condition $\mathfrak{m}^k=0$, then I think the spec of $A$ can ... | https://mathoverflow.net/users/4721 | Geometric meaning of small extensions ? | If you think of elements in a local Artin $k$-algebra $R$ as, say functions on the origin of $k^n$ which remember some (finite amount of) higher order information in the various $n$ directions, then a small extension $R'$ of $R$ is just another such ring with functions that remember "at most one order higher".
For ex... | 6 | https://mathoverflow.net/users/332 | 53633 | 33,532 |
https://mathoverflow.net/questions/53047 | 34 | There are nonnegative polynomials that are not sums of squares. For example Motzkin gave the example $x^4y^2+x^2y^4+z^6-3x^2y^2z^2$ in 1967.
Is there a real polynomial $f\in{\mathbb{R}}[x\_1,\dotsc,x\_n]$ in several indeterminates that is not a sum of squares but $f^N$ is a sum of squares for some odd integer $N>0$?
... | https://mathoverflow.net/users/nan | When is a power of a nonnegative polynomial a sum of squares? | Motzkin's original proof shows that $x^4y^2 + x^2y^4 + z^6 - a x^2y^2z^2$ is psd and
not sos for any $a$ in the interval $(0,3]$. If you take $a = .02$ say, it is reasonably
simple, though messy, to show that $(x^4y^2 + x^2y^4 + z^6 - .02x^2y^2z^2)^3$ is a
sum of squares; in fact, it's a sum of binomial squares $(x^b ... | 17 | https://mathoverflow.net/users/11935 | 53653 | 33,545 |
https://mathoverflow.net/questions/40507 | 36 | An easy consequence of the Erdős-Dushnik-Miller theorem $\kappa\to(\kappa,\omega)^2$ is the following, that will denote $(\*)$ (it appears as an exercise in Kunen's book, it was probably mentioned explicitly earlier by Dushnik-Miller or Erdős, but I haven't found a reference):
>
> If $X$ is infinite and $<\_1$ and ... | https://mathoverflow.net/users/6085 | Distinct well-orderings of the same set | Clinton Conley has found a nice argument that solves the problem. I daresay that even in the presence of choice, it is nicer than the standard approach, and avoids having to separate the argument by cases depending on the cofinality of the cardinal in question. Clinton and I are curious whether the "König lemma"-like s... | 11 | https://mathoverflow.net/users/6085 | 53659 | 33,550 |
https://mathoverflow.net/questions/53655 | 8 | I think in priciple it's possible to consider a theory of "conformal-symplectic manifolds", in an analogous fashion as the usual conformal geometry.
To spell out the spontaneous definitions: say that two symplectic forms $\omega\_1$, $\omega\_2$ on a smooth manifold $M$ are *conformal* to each other if there is a sm... | https://mathoverflow.net/users/4721 | Conformal-symplectic geometry ? | If the manifold has dimension bigger than 2, I think the conformal class of $\omega$ is just $k\omega$ for constants $k$. Locally, by Darboux we can write $\omega = \sum\_i dq^i \wedge dp\_i$. If the dimension is greater than 2, the only way for $0 = d(f\omega) = df \wedge \omega$ is if $df = 0$.
| 13 | https://mathoverflow.net/users/4622 | 53661 | 33,552 |
https://mathoverflow.net/questions/53610 | 15 | For any integer $r \geq 2$, et $V\_r$ be the set of polynomials $Q \in {\mathbb Q}[X]$ of degree $r-1$ such that there is an algebraic number $\alpha$ of degree $r$ , such that
$Q(\alpha)$ is a conjugate of $\alpha$.
It is not hard to see that $V\_2$ consists exactly of $X$ and all the polynomials $a\_0-X$, for $a\_0... | https://mathoverflow.net/users/2389 | Which polynomials arise as formulas for a conjugate | $\newcommand\Z{\mathbf{Z}}$
$\newcommand\Q{\mathbf{Q}}$
If we replace $\alpha$ by $\alpha + \lambda$ for $\lambda \in \Q$ (translation), we may
replace $Q(X)$ by $Q(X - \lambda) + \lambda$.
Similarly, if we replace $\alpha$ by $\mu \cdot \alpha$ (dilation), we may replace $Q(X)$ by
$\mu \cdot Q(X/\mu)$.
Let's dis... | 19 | https://mathoverflow.net/users/nan | 53664 | 33,555 |
https://mathoverflow.net/questions/53563 | 14 | I'm trying to follow the explanation given in Olsson's "[Sheaves on Artin stacks](http://math.berkeley.edu/~molsson/qcohrevised.pdf)" for the lack of functoriality for lisse-étale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor $f^{-1}$ sends $\def\liset{\text{lis-ét}}N \in X\_\liset$ to the sh... | https://mathoverflow.net/users/19943 | Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology? | As noted in the comments this has nothing to do with stacks.
Let F and G be a pair of adjoint functors between categories C and D (with F the left adjoint). Denote by AB(C) and AB(D) the categories of abelian group objects of C and D. Then it doesn't automatically follow that F induces a morphism from AB(C) to AB(D)... | 4 | https://mathoverflow.net/users/2 | 53666 | 33,557 |
https://mathoverflow.net/questions/53640 | 9 | Generally speaking, I am interested in counting the number of $\mathbb{F}\_p$- isomorphism classes of elliptic curves containing a specific torsion subgroup, and I was wondering if there were any simple formulas for particular torsion subgroups. Any information on this topic is appreciated.
More specifically, I would... | https://mathoverflow.net/users/9769 | Counting isomorphism classes of elliptic curves with specific torsion | $\newcommand\F{\mathbf{F}}$
$\newcommand\Z{\mathbf{Z}}$
$\newcommand\SL{\mathrm{SL}}$
Because of the existence of the Weil paring,
elliptic curves with such a subgroup only exist when
$p \equiv 1 \mod \ N$.
Let $S\_N$ denote the set of elliptic curves over $\F\_p$ such
that $E[N]$ is defined over $\F\_p$.
It will... | 13 | https://mathoverflow.net/users/nan | 53667 | 33,558 |
https://mathoverflow.net/questions/53669 | 19 | Let $a\_1,\ldots,a\_n$ be real numbers such that $\sum\_i a\_i^2 =1$ and let $X\_1,\ldots,X\_n$ be independent, uniformly distributed, Bernoulli $\pm 1$ random variables. Define the random variable
$S:= \sum\_i X\_i a\_i $
Are there absolute constants $\epsilon >0$ and $\delta<1$ such that for every $a\_1,\ldots,a\... | https://mathoverflow.net/users/12595 | Anti-concentration of Bernoulli sums | The answer to your amended question is yes. In fact, for any $\epsilon\in[0,1)$ we have
$$
\mathbb{P}(\vert S\vert > \epsilon)\ge (1-\epsilon^2)^2/3.
$$
So, we can take $\delta = 1-(1-\epsilon^2)^2/3$. This is the $L^0$ version of the [Khintchine inequality](http://en.wikipedia.org/w/index.php?title=Khintchine_inequali... | 19 | https://mathoverflow.net/users/1004 | 53683 | 33,563 |
https://mathoverflow.net/questions/53676 | 5 | I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix construction of such a representation? Are there any integral ones? Maybe in higher degrees?
| https://mathoverflow.net/users/2164 | explicit linear representations of fundamental groups of surfaces | Yes, this has been done.
MR1292919 (96f:30045)
Maskit, Bernard(1-SUNYS)
Explicit matrices for Fuchsian groups. (English summary) The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), 451–466,
Contemp. Math., 169, Amer. Math. Soc., Providence, RI, 1994.
You can fi... | 4 | https://mathoverflow.net/users/11142 | 53684 | 33,564 |
https://mathoverflow.net/questions/53641 | 5 | This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ray of light) rolls inside, going in straight lines, and upon collision with the wall, the orbit is reflected.
It is in... | https://mathoverflow.net/users/4129 | Dense orbits in billiards | Do you mean to ask whether the trajectories in almost all cases (in {shapes X trajectories}
are dense in the set of {positions, directions} on the table, or just in the set positions?
The first question seems more natural to me; the answer is *no*:
If there are
two convex portions of the boundary curve pointing toward... | 7 | https://mathoverflow.net/users/9062 | 53689 | 33,567 |
https://mathoverflow.net/questions/53673 | 14 | Gödel's [original proof](http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf) of the First Incompleteness theorem relies on [Gödel numbering](http://en.wikipedia.org/wiki/G%25C3%25B6del_numbering).
Now, the use of Gödel numbering relies on the fact that the [Fundamental Theorem of Arithmetic](http://e... | https://mathoverflow.net/users/12597 | Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering? | Just like most mathematical theorems, you can formalize Godel's Theorems in some first order language (with some "standard" interpretation under which the formalization means what it's supposed to mean), turn the proof into a purely syntactic string of formulas, and figure out which formulas in that first order languag... | 9 | https://mathoverflow.net/users/7521 | 53690 | 33,568 |
https://mathoverflow.net/questions/53711 | 5 | While reading a recent paper by Kunen [arxiv.org/abs/0912.3733](http://arxiv.org/abs/0912.3733), which deals with PFA and the existence of certain differentiable functions, (defined on all of $\mathbb{R}$) which map certain $\aleph\_1$-dense subsets of $\mathbb{R}$ onto other $\aleph\_1$-dense subsets of $\mathbb{R}$. ... | https://mathoverflow.net/users/8843 | $C^n$ And Forcing: Reading a Recent Paper By Kunen | Perhaps you could clarify what kind of strange behavior you are seeking? In terms of ordinary behavior, there are a few easy observations:
* Any forcing notion that adds a new real number will add many new $C^\infty$ functions, such as new lines and polynomials.
* If a forcing notion does not add a new real number, t... | 7 | https://mathoverflow.net/users/1946 | 53714 | 33,583 |
https://mathoverflow.net/questions/53712 | 7 | Conjecturally, every finite group is the Galois group of some extension of the rationals.
[This](https://mathoverflow.net/questions/53530/maximal-non-abelian-extensions-of-number-fields-unramified-everywhere) question made me wonder what is known about infinite
simple groups occurring as Galois groups.
*What are th... | https://mathoverflow.net/users/3503 | Infinite simple Galois groups | Any profinite simple group is finite, since it has nontrivial finite quotients (the conjugates of a finite index subgroup intersect in a finite index subgroup).
| 15 | https://mathoverflow.net/users/6451 | 53716 | 33,584 |
https://mathoverflow.net/questions/53724 | 45 | Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly approximable by rationals. But I wonder if there is another sense in which one number is more irrational than another.
Consi... | https://mathoverflow.net/users/175 | Are some numbers more irrational than others? | Yes, there is such a thing as the [**irrationality measure**](http://en.wikipedia.org/wiki/Liouville_number#Irrationality_measure) of a real number (I'm not sure if it can be / has already been extended to complex numbers). It is based on the idea that all algebraic numbers (including the golden ratio) are hard to appr... | 52 | https://mathoverflow.net/users/1916 | 53725 | 33,590 |
https://mathoverflow.net/questions/53731 | 0 | Are braid links proper links? Or are the concepts involved unrelated?
| https://mathoverflow.net/users/12606 | Are braid links proper links? | According to the definitions in your comment, the closure of the 2 stranded braid with braid word $\sigma\_1^6$ is not proper, since the closure is a 2 component link with linking number 3.
It's hard to think of a more straightforward definition than what murakami says, but if you want examples, any link with all pai... | 2 | https://mathoverflow.net/users/3874 | 53761 | 33,615 |
https://mathoverflow.net/questions/53741 | 2 | Let $\sigma\_{11}(n)$ denote the sum of the 11th powers of the positive integral divisors of the positive integer n.
Let $\tau(n)$ denote Ramanujan's tau function, which is the coefficient of $q^n$ in the power series $q \Pi\_{n=1}^{\infty} (1-q^{n})^{24}$.
It was proven by Hecke that if $n$ is a positive integer, the ... | https://mathoverflow.net/users/12610 | Proving Congruence Without Leech Lattice | Some historical comments should not be out of place. I'm writing them without checking the facts.
Ramanujan published his paper in 1916 under the modest title *On some arithmetical functions*. He made a number of conjectures about the $\tau$-function such as $\tau(mn)=\tau(m)\tau(n)$ whenever $\gcd(m,n)=1$, and a rec... | 5 | https://mathoverflow.net/users/2821 | 53768 | 33,617 |
https://mathoverflow.net/questions/53697 | 5 | Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied?
$$ f( \left(\int\_{0}^{t} g(x) \ \text{d}x\right)) = g( \left(\int\_{0}^{t} f(x) \ \text{d}x\right)) $$
**P.S.:** I migrated ... | https://mathoverflow.net/users/6770 | Are there functions satisfying the following integral condition? | Take, e.g., two (distinct, non-trivial) bump functions $F$ and $G$ s.t. $supp\: F\cap G\left(\mathbb{R}\right)=supp\: G\cap F\left(\mathbb{R}\right)=\emptyset$
. Then their derivatives $f=F^{\prime}$, and $g=G^{\prime}$ are clearly satisfying the required identity.
| 9 | https://mathoverflow.net/users/2508 | 53775 | 33,620 |
https://mathoverflow.net/questions/53778 | 2 | I was trying to prove that $ Aut( S\_g $), g$ \geq 2 $ [ orientation preserving isometries ] is finite in the following way :
For fixed $M $ ( positive ) there are finitely many , say $ k $ number of simple closed geodesics ( with repeated multiplicities ), say $c\_1...c\_k $ with length $\leq M$. Consider the group ... | https://mathoverflow.net/users/6953 | Can we prove $ Aut(S_g) , g \geq 2 $ is finite in the following way ? | The answer is yes. It's a little easier if we consider oriented geodesics (which is fine for your argument). Let $\alpha$ and $\beta$ be two oriented simple closed curves on $S\_g$ that intersect once. Let $a$ and $b$ be geodesics that are homotopic to $\alpha$ and $\beta$, respectively. Then $a$ and $b$ only intersect... | 4 | https://mathoverflow.net/users/317 | 53788 | 33,627 |
https://mathoverflow.net/questions/53790 | 1 | I want to understand the geometry of the Hyperbolic annuli (Hyperbolic plane quoteinted by the group generated by $z\mapsto rz$ for a fixed $r$) and parabolic annuli (Hyperbolic plane quoteinted by the group generated by $z\mapsto z + 1$). Some questions i want to understand are :Injectivity radius at an arbitrary poin... | https://mathoverflow.net/users/6822 | Good references for Hyperbolic and parabolic annuli | Alan Beardon's "Geometry of Discrete Groups" should give you all the tools you need to answer your questions (the tools being computation of distances and angles in a convenient model).
| 2 | https://mathoverflow.net/users/11142 | 53792 | 33,629 |
https://mathoverflow.net/questions/53787 | 4 | It is easy to see that there cannot be a proper holomorphic map from the punctured unit disk to the unit disk in the complex plane. What about the other direction; does there exist a proper holomorphic map from the unit disk to the punctured unit disk?
| https://mathoverflow.net/users/36038 | Proper holomorphic map from unit disk to punctured unit disk | I don't think so. A map $\mathbb D \to \mathbb D^\ast$ would lift to a map $\mathbb D\to \mathbb H$, where the upper halfplane $\mathbb H$ is seen as the universal cover of $\mathbb D^\ast$. As $\mathbb D \to \mathbb D^\ast$ is proper, so is $\mathbb D\to \mathbb H$. In particular it has closed image but by the open ma... | 10 | https://mathoverflow.net/users/4008 | 53794 | 33,630 |
https://mathoverflow.net/questions/53782 | 13 | I'm wondering about the theoretical placement of quasifibrations.
One nice thing about "weak fibrations" (maps homotopy equivalent in the category of maps to Hurewicz fibrations) is that a pullback square involving (one) weak fibration is a homotopy pullback square.
Is the corresponding result
true for quasifibr... | https://mathoverflow.net/users/3634 | Quasifibrations and homotopy pullbacks | The definition of quasifibration (according to Dold & Thom, 1958) is: a map $f:E\to B$ such that for all $b$ in $B$, the canonical map from the fiber to the homotopy fiber is a weak equivalence. Pullbacks with respect to such maps are *not* generally homotopy pullbacks; an example was given in that 1958 paper (Bermerku... | 20 | https://mathoverflow.net/users/437 | 53799 | 33,633 |
https://mathoverflow.net/questions/53803 | 15 | A group $G$ is said to be *elementary amenable* if it can be obtained from finite and abelian groups by subgroups, quotients, extensions and increasing unions. It is well-known that all such groups are amenable, i.e. allow for a finitely additive and $G$-invariant probability measure on $G$. [Grigorchuk's group](http:/... | https://mathoverflow.net/users/8176 | Amenable groups with finite classifying space | There are very few examples of amenable but not elementary amenable finitely presented groups. There exists a torsion-free example, an ascending HNN extension of the basilica group
(Bartholdi, Laurent; Virág, Bálint, Amenability via random walks.
Duke Math. J. 130 (2005), no. 1, 39–56 - there it is proved that the basi... | 8 | https://mathoverflow.net/users/nan | 53812 | 33,642 |
https://mathoverflow.net/questions/52913 | 17 | Consider the following two conditions for a group $G$:
(1) $G$ does not satisfy a nontrivial law.
(2) $G$ contains a non-abelian free subgroup.
Obviously (2) implies (1) and it is easy to construct torsion groups that do not satisfy any law (e.g., the direct product of all finite groups). Thus (1) does not imply... | https://mathoverflow.net/users/10251 | Free subgroups vs law | Since Henry asked, here is a reference: Aner Shalev in the first chapter (Lie Methods in the Theory of pro-$p$ Groups) in New Horizons in pro-$p$ Groups posed 4 conjectures in decreasing order of strength:
1. Let $G$ be a finite $p$-group satisfying some identity $w$ with probability $\epsilon>0$. Then $G$ satisfies ... | 6 | https://mathoverflow.net/users/5034 | 53814 | 33,644 |
https://mathoverflow.net/questions/53816 | 2 | Let $S$ be a complex surface whose fundamental group is isomorphic to the fundamental group of a product of two curves of genera $>1$. Does $S$ have to be a product of two curves?
| https://mathoverflow.net/users/12622 | Fundamental group of a product of two curves | The answer in **no**, because of the following result:
**Theorem 1.** Let $X$ be a non-ruled minimal surface. Then there exists a finite ramified covering $S \to X$ of degree $>1$, such that $S$ is minimal of general type with $K\_S$ very ample, $\pi\_1(S) \cong \pi\_1(X)$ and $S$ is *not* birationally equivalent to ... | 15 | https://mathoverflow.net/users/7460 | 53823 | 33,649 |
https://mathoverflow.net/questions/53824 | 7 | Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to the tangent vector of L at the cirle's center. S does not intersect itself.
What is the shortest possible length of L... | https://mathoverflow.net/users/12249 | Length of shortest possible knot | The invariant you are talking about is usually called the "ropelength" of the knot. You can find some basic stuff at the wikpedia page <http://en.wikipedia.org/wiki/Ropelength> which also gives some good references. (Note that some people use unit circles, while other people use circles of diameter 1, so the reported r... | 19 | https://mathoverflow.net/users/4087 | 53828 | 33,651 |
https://mathoverflow.net/questions/53841 | 7 | Suppose you are given a domain $\Omega \subset \mathbb{R}^n,$ and a (Morse) function $f: \Omega \rightarrow \mathbb{R},$ all of whose critical points are positive-definite.
The question is: is there a diffeomorphism $\phi: \Omega \rightarrow \Omega$ such that $f \circ \phi$ is a *convex* function?
**Another edit** Th... | https://mathoverflow.net/users/11142 | Hidden convexity |
>
> Edited in response to edits of the question. Original answer follows new answer.
>
>
>
If you don't require that $f$ is proper, then there are many counterexamples already
in $\mathbb R^2$ (see below, in the answer to question as previously worded).
If you require that $f$ is a **proper** Morse function (t... | 6 | https://mathoverflow.net/users/9062 | 53848 | 33,660 |
https://mathoverflow.net/questions/53839 | 5 | I want to be clear about my phrasing, but I'm not a topologist, so when I say "knot" or "link" I mean the equivalence class under ambient isotopy of an embedding of the circle into $\mathbb{R}^3$.
For a while I have been looking for references to what I've taken to calling the "Fraïssé link" - this thing would have (... | https://mathoverflow.net/users/4594 | What is known about links with a countably-infinite number of tame components? | I don't think that knot theorists are going to be very interested in such infinite links, but they do occur sometimes in the wider area of geometric topology, for instance in the proof of theorem 1.1 [here](https://arxiv.org/abs/0812.1407). Still I doubt that they have been studied per se.
My understanding is that yo... | 2 | https://mathoverflow.net/users/10819 | 53851 | 33,663 |
https://mathoverflow.net/questions/53822 | 6 | Introduction
------------
Suppose we are trying to prove that $\rm PSO\_3\times PSO\_3$ is isomorphic with $\rm PSO\_4,$ and we catch on to the idea of using the quaternions to do so. We realize (as in Conway & Smith's *On Quaternions and Octonions,* whence the quotation) that we can encapsulate $\rm PSO\_3$ as the s... | https://mathoverflow.net/users/9793 | In the quaternions, "any imaginary unit may be called i" | Just to elaborate on what is already in the comments, the algebra automorphisms of $\mathbb H$ act transitively on the set of pairs $(u,v)$ where $u$ and $v$ are imaginary quaternions of unit length that are orthogonal to one another.
To see this, I will include here some remarks on $\mathbb H$ and its automorphisms... | 20 | https://mathoverflow.net/users/2874 | 53856 | 33,664 |
https://mathoverflow.net/questions/53854 | 4 | Does anyone know the name/author of the model theory paper where it's proven that you can define the $ < $ relation on $ (\mathbb{Q}, +, \cdot, 1, 0) $?
| https://mathoverflow.net/users/12631 | "Less than" formula for complete theory of the rationals | Every natural number is the [sum of four squares](http://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem), and so you
can define the positive rational numbers as those of the
form $(a^2+b^2+c^2+d^2)/(e^2+f^2+g^2+h^2)$, where the denominator is not zero, and this is
expressible in your language. And the order is ... | 12 | https://mathoverflow.net/users/1946 | 53857 | 33,665 |
https://mathoverflow.net/questions/53855 | 25 | Suppose that $\epsilon\_1,\epsilon\_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon\_n=\pm1)=1/2$, and $a\_1,a\_2,\ldots$ is a real sequence with $\sum\_na\_n^2=1$. Letting $S=\sum\_n\epsilon\_na\_n$, the question is whether there exists a constant $c > 0$, independent of the cho... | https://mathoverflow.net/users/1004 | An $L^0$ Khintchine inequality | OK. Here's a proof that $c > 0.002$. No doubt it can be substantially improved. We can assume the $a\_i$ are arranged in decreasing order. Write $a$ for $a\_1$.
If $a\ge 1/2$, let $X=a\_1\epsilon\_1$ and $Y=(1-a^2)^{-1/2}(a\_2\epsilon\_2+\ldots+a\_n\epsilon\_n)$ so that $S=X+\sqrt{1-a^2}Y$. Notice that $Y$ is of the ... | 23 | https://mathoverflow.net/users/11054 | 53869 | 33,673 |
https://mathoverflow.net/questions/53865 | 1 | Let's put $m>n$ two nonnegative integers and $Gr:=Grass(n,k^m)$ the grassmanian of the subspaces of dimension $n$ in $k^m$. We have a natural immersion $Gr \subset P({\Lambda}^{n} k^m)$ and I call $C(Gr) \subset {\Lambda}^{n} k^m$ the affine cone above $Gr$.
If we blow-up $C(Gr)$, we always obtain something smooth, bu... | https://mathoverflow.net/users/23236 | crepant resolution | The result of the blowup is the total space of line bundle $O(-1)$ on the Grassmannian $Gr$. It follows that its canonical class equals the canonical class of the Grasmannian (i.e. $-m$) plus the relative canonical class of the total space (i.e. $1$). Thus the canonical class is $1-m$ which as you see is always negativ... | 3 | https://mathoverflow.net/users/4428 | 53874 | 33,676 |
https://mathoverflow.net/questions/53872 | 25 | I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.
>
> What is the vertex angle of a regular $n$-simplex?
>
>
>
Background: For a vertex $v$ in a convex polyhedron $P$, the vertex angle at $v$ is the proportion of the volume that $P$ occupies in a... | https://mathoverflow.net/users/806 | Angle of a regular simplex | In the paper by John Leech,
"Sphere packings in Higher Space"
*Canadian Journal of Mathematics*, 1964, which you can find at the [Google book links here](https://books.google.com/books?id=ZgwT9xyHZ4gC&pg=PA675&lpg=PA675&dq=%25252522regular+simplex%25252522+solid+angle+at+vertex&source=bl&ots=Yx9JO0VskH&sig=YNEavV1sBX1j... | 20 | https://mathoverflow.net/users/6094 | 53881 | 33,679 |
https://mathoverflow.net/questions/53885 | 1 | Hi this Question follows after the answer of [Douglas Zare](https://mathoverflow.net/questions/12327/extension-of-some-feature-of-sde-ornstein-uhlenbeck-type) to this post :
So let's be given a positive function $V(x)$ which is smooth enough to have Lispchitzian derivative at all point. Moreover this function admits ... | https://mathoverflow.net/users/2642 | Follow up question, Ornstein-Uhlenbeck Extension with n mean-reversion values | Let us assume you are interested in the solution of $dX\_t=\sigma(X\_t)dW\_t+b(X\_t)dt$. If the so-called *speed measure* $m$ has finite mass, the diffusion converges in distribution to $m/|m|$. Here, $m(x)=2/(\sigma^2(x)s'(x))$ where $s$ is the so-called *scale* of the diffusion. Recall that $s$ is uniquely defined, u... | 5 | https://mathoverflow.net/users/4661 | 53889 | 33,684 |
https://mathoverflow.net/questions/53876 | 11 | Consider an open subset $U \subseteq \mathbb{R}^n$ and a smooth function $f\colon U \longrightarrow \mathbb{R}$ with $f(x) \ge 0$ for all $x \in U$.
It is then known (if I remember correctly: by Michor?) that $f = g^2$ with a function $g$ which can be shown to be twice differentiable but not $C^2$ in general. In part... | https://mathoverflow.net/users/12482 | Hilbert's 17th Problem for smooth functions | No. Let $f=z^6 + x^4 y^2 + x^2 y^4 − 3x^2 y^2 z^2$. By the [AM-GM inequality](http://en.wikipedia.org/wiki/AMGM), $f$ is nonnegative.
Suppose that $f=\sum g\_i^2$, with the $g\_i$ smooth. Expand each $g\_i$ in a Taylor series around $0$: $g\_i = a\_i + b\_i(x,y,z) + c\_i(x,y,z) + d\_i(x,y,z) + O(|x|+|y|+|z|)^4$, with... | 17 | https://mathoverflow.net/users/297 | 53893 | 33,687 |
https://mathoverflow.net/questions/53883 | 0 | Given a smooth projective surface, I consider vector bundles of rank $r$ with fixed Harder-Narasimhan filtration. Does there exist a moduli space of such bundles? If yes, how is it constructed and which are its properties (dimension, smoothness....)?
| https://mathoverflow.net/users/33841 | Moduli space of vector bundles with fixed Harder-Narasimhan filtration? | Dear ginevra,
your question is related to my earlier question [Moduli of Extensions](https://mathoverflow.net/questions/25484/moduli-of-extensions).
There are some obvious problems which occure already if you fix the HN-factors themselves:
Let $E,F$ be sheaves with Ext^1(E,F) one-dimensional. Then there are two iso... | 2 | https://mathoverflow.net/users/5714 | 53897 | 33,690 |
https://mathoverflow.net/questions/53867 | 0 | I am interested in the efficient computability of sequences.
Is it possible some ``interesting sequences'' be computed via addition formulae/semigroup operation?
Here is an artificial example.
Suppose one finds an associative operation $f : \mathbb{Z}^2 \times \mathbb{Z}^2 \to \mathbb{Z}^2$ and a sequence $a\_n$ ... | https://mathoverflow.net/users/12481 | Sequences, semigroups, addition formulae. | Is your functional equation a non-vacuous constraint? Since summation is group operation, the pairs [2n,2n+1] form a group 2N, which map to your group [a(2n),a(2n+1)]. For example, for A000069
```
1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, ...
0 1 2 3 4 5 6 7 8 9... | 1 | https://mathoverflow.net/users/11632 | 53907 | 33,694 |
https://mathoverflow.net/questions/53903 | 5 | More specifically, with $I=[0,1]$ let $E=(X,\mathcal T\ )=C^\infty(I)$, where $X$ is the underlying (say real) vector space and $\mathcal T\ $ is the (standard projective limit) topology of uniform convergence of each fixed derivative. One asks whether there is a $\mathcal T\ $−closed infinite−dimensional vector subspa... | https://mathoverflow.net/users/12643 | Is there an infinite−dimensional Banach subspace in C^∞([0,1]) ? | This was a many years lasting problem to me, but now that I began to think of it anew, I found the solution: Since $E$ has the Heine−Borel property, taking $V$ to be $\mathcal T\ $−closed, we get $S\cap V$ also such, and hence $\mathcal T\ $−compact. Having a compact zero neighbourhood, so $S$ must be finite−dimensiona... | 0 | https://mathoverflow.net/users/12643 | 53917 | 33,697 |
https://mathoverflow.net/questions/53923 | 14 | There are many versions of the Baum-Connes conjecture (the original, coarse, with coefficients, etc.). I would like to know what group theory results are needed in order to prove or disprove one of these conjectures.
| https://mathoverflow.net/users/nan | Baum-Connes conjecture | This January in New Orleans Paul Baum gave a rather extensive survey of the history and status of Baum/Connes, so I am guessing that he has a historical survey written up, or quasi-written up, since I am not seeing it on his web page), so I would strongly suggest just asking him. Sadly, I don't believe he is a MO parti... | 5 | https://mathoverflow.net/users/11142 | 53925 | 33,699 |
https://mathoverflow.net/questions/53919 | 9 | Hi
Can one explain to me what is the Hochschild homology of Fukaya category?
I mean the definition.
You can use the notations of FOOO (Fukaya-Oh-Ono-Ohta) if it helps you to explain easier.
I know what the Fukaya category is but I am very poor when it comes to algebra.
Also please explain what is the correspondin... | https://mathoverflow.net/users/5259 | Hochschild homology of Fukaya category in mirror symmetry | As Kevin comments, Hochschild homology and cohomology are defined for any $A\_\infty$-category $\mathcal{A}$. That includes Fukaya categories of symplectic manifolds and dg enhancements of the bounded derived category of varieties.
The most concrete definition of Hochschild homology $HH\_\ast(\mathcal{A},\mathcal{A})... | 15 | https://mathoverflow.net/users/2356 | 53947 | 33,710 |
https://mathoverflow.net/questions/53930 | 4 | I'm looking for information on how to compute the distribution of the random vector
$$Z = \int\_0^t f(B\_s) ds$$
where $t>0$ is fixed, $B\_s$ is a 2D Brownian bridge with $B\_0 = 0$, $B\_t=b \in \mathbb{R}^2$, and $f : \mathbb{R}^2 \rightarrow \mathbb{R}^K$ has components $f\_k : \mathbb{R}^2 \rightarrow \mathbb{R}... | https://mathoverflow.net/users/3936 | Time-integral of a smooth, vector-valued function of a planar Brownian bridge | To compute the *distribution* of $Z$ for a general function $f$ might be difficult. To evaluate it, one could rewrite the process $(B\_s)\_{0\le s\le t}$ as $B\_s=W\_s+(b-W\_t)(s/t)$ for every $0\le s\le t$, where $(W\_s)\_{s\ge0}$ is a standard 2D Brownian motion starting from $W\_0=0$.
| 1 | https://mathoverflow.net/users/4661 | 53957 | 33,715 |
https://mathoverflow.net/questions/53901 | 9 | This question concerns alternative characterizations of free group factors. The [ping pong lemma](http://en.wikipedia.org/wiki/Ping-pong_lemma) is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like criterion that can be used to determine if a given type $II\_{1}$ factor ... | https://mathoverflow.net/users/6269 | Ping Pong and Free Group Factors | I am guessing that the answer is "yes" if you interpret the question in the following way. Let $A\_i$ be some subalgebras of a von Neumann algebra $(M,\tau)$ and assume that there are mutually orthogonal Hilbert subspaces $H\_i$ of $H=L^2(M)$ so that for all $i$, $x (H\ominus H\_i) \subset H\_i$ whenever $x\in A\_i$ wi... | 12 | https://mathoverflow.net/users/12660 | 53958 | 33,716 |
https://mathoverflow.net/questions/53960 | 2 | For past months I've been trying to estimate associated Legendre function $P\_{-\frac{1}{2}+it}^m (\cosh r)$ in order to study Laplacian eigenfunctions on a hyperbolic surface. I found reasonably sharp upper bound of $C\_m(t,r):=m!P\_{-\frac{1}{2}+it}^{-m} (\cosh r)$ for non-negative $m$ when,
1) $r$ is fixed, $t$ is... | https://mathoverflow.net/users/12662 | Estimating associated Legendre function | I randomly (luckily) found this link <http://dlmf.nist.gov/14.26> just now. I hope I can find desired answer from here. Thanks for viewing my question and still, I will appreciate any advise.
| 2 | https://mathoverflow.net/users/12662 | 53961 | 33,717 |
https://mathoverflow.net/questions/53800 | 9 | Is there any book explaining in detail the book "Basic Number Theory" by Andre Weil as Dirichlet did to "Disquisitiones Arithmeticae"?
This is because I have read the two books mentioned above and I hope there will be one.
| https://mathoverflow.net/users/11059 | Is there any book explaining in detail the book "Basic Number Theory" by André Weil as Dirichlet did to "Disquisitiones Arithmeticae" by Gauss? | Indeed Ramakrishnan and Valenza 's book is a pretty good reference.
Perhaps we could give more specific answers if you were more precise about exactly where your difficulties are?
EDIT: Since we've been given precisions in the comments below, I can confirm R&V's book will nicely do for the basics of the theory ; to... | 7 | https://mathoverflow.net/users/12664 | 53972 | 33,726 |
https://mathoverflow.net/questions/53971 | 36 | I was explaining to my students that if there is an inequality between two norms, then there is an inclusion between their spaces of convergent sequences, *with matching limits*. I then proceeded to show examples of such inequalities on the normed spaces they knew, and counterexamples of sequences which converge for a ... | https://mathoverflow.net/users/12664 | Example of sequences with different limits for two norms | Consider the space $X$ of trigonometric polynomials (with period $1$, say). Choose the norms
$$\|f\|\_1=\sup\{|f(x)|;\frac16\le x\le\frac13\},\qquad \|f\|\_2=\sup\{|f(x)|;\frac23\le x\le\frac56\}.$$
Now consider the partial sums $f\_N$ of the Fourier series of the periodic function $F$ defined by $F(x)=0$ if $x\in(0,1/... | 19 | https://mathoverflow.net/users/8799 | 53981 | 33,731 |
https://mathoverflow.net/questions/53955 | 12 | If $\Gamma$ is a compact simply-connected semisimple Lie group, then the Weyl Dimension Formula tells us exactly which dimensions it can act irreducibly on.
For certain $\Gamma$, it is easy to find pairs of nonisomorphic representations of the same dimension:
1). $A\_n (n\geq 2)$, $C\_n$ ($n$ = [A116940(k)](http://... | https://mathoverflow.net/users/12301 | Which compact groups have nonisomorphic irreducible representations of the same dimension? | We give a table for irreducible representations up to dimension $2^{15}$ in the supplement of our preprint <http://arxiv.org/abs/1012.5256v1>
I immediately found the following examples with respect to the first question:
(1) $B\_3$ occurs twice in dimension 112, 168, etc.
(2) $B\_4$ occurs three times in dimension ... | 8 | https://mathoverflow.net/users/9666 | 53982 | 33,732 |
https://mathoverflow.net/questions/53989 | 7 | The broad, generic and badly posed question may be formulated in this way:
>
> Let $X$ be a compact Kälher manifold (or even a projective one). If one considers the Hodge decomposition $H^k(X, \mathbb{C}) = \bigoplus\_{p+q=k} H^{p,q}(X)$ and interprets the left hand side as Betti cohomology $\text{Hom}(\pi\_1(X), \... | https://mathoverflow.net/users/12325 | Hodge decomposition in Betti cohomology | This is really a comment on Donu Arapura's answer, but it seemed large enough to deserve it's own post. Working again in the case of $GL\_1$, Simpson considers three spaces:
$M\_{betti}$: The space of $\mathbb{C}^\*$-local systems on $X$. If you like, you can think of this as smooth complex line bundles with an integ... | 3 | https://mathoverflow.net/users/297 | 54000 | 33,744 |
https://mathoverflow.net/questions/53999 | 23 | I can't seem to find any work on the following question: Can every (closed, of finite type) Riemann surface $S$ be realized as an embedded (or even immersed) smooth surface in Euclidean $3$-space, where by realized, I mean that the induced conformal structure on $T \subset \mathbb{E}^3$ is the conformal structure on $S... | https://mathoverflow.net/users/11142 | Conformal embedding of Riemann surfaces into 3-space | The embedding question has been answered in the affirmative by Adriano Garsia in 1961. Here is the [link](http://www.ams.org/mathscinet-getitem?mr=125591) to MR. In fact, he has [shortly after shown](http://www.ams.org/mathscinet-getitem?mr=148898) that the embedding might be chosen with real algebraic image.
| 24 | https://mathoverflow.net/users/6451 | 54001 | 33,745 |
https://mathoverflow.net/questions/53990 | 13 | The goal of this question is to find a "geometric" definition of Frobenius element in $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
Here are two definitions that don't work, but that should help explain what I mean. Fix an algebraic closure of $\mathbb{Q}$ and, for a prime $p$, fix an algebraic closure of $\mathbb{... | https://mathoverflow.net/users/290 | Frobenius elements from the point of view of étale fundamental groups | Your guess is right. $\pi\_1(Spec(R))$ is the automorphism group of the maximal extension $\mathbb{Q}^{p\textrm{-}ur} \subset \overline{\mathbb{Q}}$ unramified at $p$. (This is a special case of SGA 1 Exp. V Prop 8.2 about the fundamental group of normal schemes). The morphism $g\_\*$ is simply the restriction map $Gal... | 6 | https://mathoverflow.net/users/1985 | 54013 | 33,752 |
https://mathoverflow.net/questions/54033 | 12 | The following question came up in a discussion with my advisor:
>
> Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-algebra of $\mathcal F$. Does there exist an event $U \in \mathcal F$ such that
>
>
> * $U$ is independent of $\mathc... | https://mathoverflow.net/users/238 | Does there exist an event independent of a given sigma-algebra? | No. For a very simple example, take $\Omega = \{a,b,c\}$ consisting of three points, with $\mathcal{F} = 2^\Omega$ and $P(A) = |A|/3$ the uniform measure. Let $\mathcal{G} = \{\{a\}, \{b,c\}, \Omega, \emptyset\}$. Then $\mathcal{G}$ is a proper sub-$\sigma$-algebra but no nontrivial event in $\mathcal{F}$ is independen... | 15 | https://mathoverflow.net/users/4832 | 54034 | 33,764 |
https://mathoverflow.net/questions/53937 | 8 | Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End\_{\mathbb{C}}(A) \otimes\_{\mathbb{Z}} \mathbb{Q}$. (So $n$ is a multiple of $g$.)
Then, the tangent space to the identity of $A$ defines an $n$-dimensional complex rep... | https://mathoverflow.net/users/12087 | possible CM-types of abelian varieties | There are certain additional conditions on ``multiplicities" that are spelled out in the original Shimura's paper. Let me discuss a couple of interesting cases.
The case of CM type, i.e. when $n=g$. Let $\Phi$ be the corresponding CM type, which is a $n$-element set of embeddings of $K$ into the field $C$ of complex ... | 5 | https://mathoverflow.net/users/9658 | 54036 | 33,766 |
https://mathoverflow.net/questions/54039 | 2 | I am looking for an "easy" proof of the following statement: Suppose that $X$ is a simply connected space for which $\pi\_2(X)=0$. Then $H\_2(X)=0$ as well.
I know that one can use the Hurewicz theorem to prove this, but I feel like there must be a simpler proof.
| https://mathoverflow.net/users/10269 | vanishing of $\pi_2$ and $H_2$ | Claim 1: If $x\in H\_2X$ is a homology class, then there exists a 2-dimensional CW complex $K$ and a map $f:K\to X$, such that $x$ is in the image of $f\_\*:H\_2K\to H\_2X$.
Claim 2: If $X$ is simply connected with $\pi\_2X=0$, then every map $K\to X$ from a 2-dimensional CW complex is null homotopic.
Claim 1 can b... | 11 | https://mathoverflow.net/users/437 | 54043 | 33,769 |
https://mathoverflow.net/questions/54028 | 8 | I'll begin by saying that I'm not sure what I want to ask specifically, but pretty sure what in general, so please don't hold my misunderstandings against me, but do comment on them.
I know that the unit group of a number field is finitely generated, and so is $SL\_2(\mathbb{Z})$. I understand that so are $SL\_n(\mat... | https://mathoverflow.net/users/2024 | Are affine groups over rings of integers finitely generated? | It seems to me that if you are thinking of affine groups, then the appropriate result is that $S$-arithmetic subgroups of reductive linear algebraic groups over number fields are finitely generated. Over function fields there are exceptions (which I think are known explicitly).
This includes examples such as $SL\_2(... | 9 | https://mathoverflow.net/users/10849 | 54044 | 33,770 |
https://mathoverflow.net/questions/54038 | 7 | Let $G$ be a graph on $n$ vertices with $an^2$ edges containing at most $an^2/2$ copies of $K\_4$. If there are cubically many triangles, say $cn^3$, then there is at least one edge that is not contained in any $K\_4$.
Note that it is necessary to require that the number of $K\_4$'s is significantly smaller than the... | https://mathoverflow.net/users/12674 | Graphs with many triangles but few complete graphs on 4 vertices | Your statement may be true for large enough values of $c$, but it is not true for all constants $c>0$.
Specifically, for small enough values of $c$, form a counterexample $G$ consisting of the disjoint union of two subgraphs:
* $K\_{n\sqrt 2,n\sqrt 2,n\sqrt 2}$ together with one extra edge in each component of the ... | 11 | https://mathoverflow.net/users/440 | 54049 | 33,772 |
https://mathoverflow.net/questions/53809 | 9 | I it true that any connected oriented 3-manifold with nonempty boundary can be obtained as a ramified cover of 3-ball with a ramification at a *link*?
* link = a 1-dimensional submanifold with possibly nonempty boundary.
If answer is "YES", can we choose in addition the restriction of the covering at the boundary?... | https://mathoverflow.net/users/1441 | Ramified cover of 3-ball | [Berstein and Edmonds](http://www.ams.org/journals/tran/1979-247-00/S0002-9947-1979-0517687-9/home.html) prove in Cor. 6.3 that for an orientable 3-manifold $W$ with connected boundary, with a branched cover $\varphi: \partial W\to S^2$ of degree $n>3$, then there is a branched cover $\Phi: W\to D^3$ such that $\Phi\_{... | 8 | https://mathoverflow.net/users/1345 | 54057 | 33,776 |
https://mathoverflow.net/questions/54018 | 20 | I (probably re)invented a very short combinatorial proof of Wilson's theorem that I perennially teach my students. If it occurs in the literature and someone can tell me where first (or even at all), I would like to attribute credit properly.
I actually prove $p! - p(p-1) \equiv 0 \mod p^2$.
$p!$ counts bijective ... | https://mathoverflow.net/users/10909 | A particular combinatorial proof of Wilson's theorem | According to Dickson's History of the Theory of Numbers, this proof was first found by J. Petersen, Tidsskrift for Mathematik (3), 2, 1872, 64-5. (Petersen divides everything by 2, but the idea is the same.) Dickson gives a number of references to rediscoveries of Petersen's proof in the 19th century. Petersen was also... | 18 | https://mathoverflow.net/users/10744 | 54058 | 33,777 |
https://mathoverflow.net/questions/54037 | 21 | In algebraic geometry, an etale sheaf on a Noetherian scheme is called *constructible* if the scheme has a finite stratification by locally closed subschemes such that the pullback of the sheaf to each of the subschemes is lcc (locally constant constructible). This definition seems to make sense both for sheaves of abe... | https://mathoverflow.net/users/1384 | Naive question about constructing constructible sheaves. | I wonder if this may suffice for your purposes: suppose you have a scheme $X$, cut up into an open subscheme $U$ and a closed subscheme $Z$ supported on the complement of $U$ in $X$. Let $j:U\to X$ and $i:Z\to X$ be in the inclusions. Let $F$ be an \'etale sheaf of sets on $X$. One derives from $F$ a triple
$$
(j^\*F, ... | 15 | https://mathoverflow.net/users/2490 | 54062 | 33,779 |
https://mathoverflow.net/questions/54026 | 0 | Let $O$ be an open subset of the separable Hilbert space $H.$ Let $E$ be a separable Banach space. Is it true that $C^0\_b(O;E),$ the space of bounded continuous maps $O\rightarrow E$, endowed with the $C^0$-norm, is separable? If YES, where can I find I proof of this fact?
| https://mathoverflow.net/users/3509 | separability of a certain space of continuous functions | The answer is negative. For, pick some non-zero $e$ in $E$, and
choose a surjection $\rho\in C\left(O,\mathbb{R}\right)$ (there exists
!).
Next, consider the (uncountable, uniformly discrete) family of functions {
$f\_{A}$; $A\subset\mathbb{Z}$ nonempty } $\subset C\_{b}^{0}\left(O,E\right)$, expressed by $$f\_{A}\le... | 3 | https://mathoverflow.net/users/2508 | 54063 | 33,780 |
https://mathoverflow.net/questions/54059 | 2 | The angular central Gaussian (ACG) distribution on $(p-1)$-dimensional sphere $\mathbb{S}^{p-1}$ for a symmetric positive definite parameter matrix $\mathbf{A}$ is defined as
$$f(\mathbf{x},\mathbf{A}) = |\mathbf{A}|^{-1/2} (\mathbf{x}^T\mathbf{A}^{-1}\mathbf{x})^{-p/2}.$$
As many paper pointed out, the name 'angul... | https://mathoverflow.net/users/11361 | deriving angular central gaussian distribution from a multivariate normal distribution | If you integrate the Gaussian distribution function $f\_p (\mathbf{x})$ in $p$-dimensions along the radial direction from 0 to $\infty$ I believe you will derive the angular pdf you wrote down. Specifically, for $\mathbf{x} \in\mathbb{S}^{p-1}$, the integral $$\int\_0^{\infty} t^{p-1} dt \; |\mathbf{A}|^{-1/2} \exp(-\f... | 3 | https://mathoverflow.net/users/12409 | 54073 | 33,784 |
https://mathoverflow.net/questions/54076 | 8 | For an object $X$ of a category, $h\_X$ is the contravariant functor represented by $X$, i.e. $h\_X = Hom(-,X)$.
**Question** a) Who invented this notation? (My guess: Grothendieck)
b) Is there a special reason why the letter $h$ was chosen? Is it in an abbreviation for "homomorphism"?
| https://mathoverflow.net/users/2841 | Notation for a representable functor | Let me answer the questions in order.
**a**) It was invented by Grothendieck, see EGA I, Springer edition, especially chapter 0, discussion of representable functors.
**b**) Quite possibly is a shortcut for $Hom$. Sometimes the letter $y$ is used (for Yoneda). The trouble is when you are considering the representab... | 12 | https://mathoverflow.net/users/6348 | 54085 | 33,788 |
https://mathoverflow.net/questions/54080 | 1 | I need a function $f(x)$ such that given a set of values $X=\{x1, x2, ...\}$ and a function $rand()$ that returns a value between 0 and 1, $f(x)$ returns a value that increases with x such that the probability that $f(a)$ returns a value greater than $f(b)$ is based on their relative sizes: $P[f(a) \ge f(b)] = \frac{a}... | https://mathoverflow.net/users/12683 | Is there an f(x) such that P[f(a) >= f(b)] = a/(a+b) given a set of possible values for a and b? | If I interpret your question correctly, the answer is that the exponential random variable $\xi$ (density $e^{-x}$) has the property that for two independent copies $\xi\_1,\xi\_2$ of $\xi$, one has $P(a\xi\_1\le b\xi\_2)=\int\_{y\_1\le ry\_2}e^{-y\_1-y\_2}dy\_1dy\_2=\frac r{1+r}$ with $r=\frac ba$, which is exactly wh... | 5 | https://mathoverflow.net/users/1131 | 54089 | 33,790 |
https://mathoverflow.net/questions/52960 | 30 | I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau's list was published, so what is the status of this problem today?
Obviously it isn't hasn't been shown to be true, b... | https://mathoverflow.net/users/4054 | Which almost complex manifolds admit a complex structure? | In complex dimension 3 or more it is still an open conjecture
(which was re-stated Yau a couple of years ago in his UCLA lectures).
There is not a single known example of an almost complex manifold
of dimension $\geq$ 3 not admitting a complex structure.
In dimension 2 it is easy, of course, because
the non-Kähler c... | 18 | https://mathoverflow.net/users/3377 | 54094 | 33,792 |
https://mathoverflow.net/questions/54051 | 23 | In "[The maximum number of Hamiltonian paths in tournaments](https://doi.org/10.1007/BF02128667)" by Noga Alon, the author states the following without proof (equation 3.1):
"Consider a random permutation $\pi$ of $\mathbb{Z}\_n$. What is the probability that $\pi(i+1)−\pi(i) \mod{n} < n/2$ for all $i$?"
The claim ... | https://mathoverflow.net/users/6994 | Random permutations of Z_n | I emailed Noga to ask him; here is his response (touched up slightly for MO; any errors in what I post are probably mine rather than Noga's). The only details not present are the required applications of Stirling's formula.
>
> As far as I recall the argument I had in mind was as follows (I am not trying to optimiz... | 14 | https://mathoverflow.net/users/3401 | 54103 | 33,800 |
https://mathoverflow.net/questions/54111 | 3 | This is a follow-up of a question of mine with a similar title. I am interested in Morse homology (on Hilbert manifolds), more specifically with "generic" perturbations of the metric tensor (under the heading of "transversality"). The space of perturbations to use should have the property of separability in order to ap... | https://mathoverflow.net/users/3509 | separability of a certain space of continuous functions, II | **No.** Ady's construction still works, I think. Here's another, easier one:
Choose an orthonormal basis $\{e\_{n}\}$ of $H$. Since $\|e\_{i} - e\_{j}\| = \sqrt{2}$, the continuous functions $f\_{n}(x) = \max\{0, 1 - 2 \cdot \|e\_{n} - x\|\}$ have disjoint support and sup-norm $1$. This gives an isometric embedding $... | 6 | https://mathoverflow.net/users/11081 | 54114 | 33,804 |
https://mathoverflow.net/questions/54109 | 13 | Let h\* be a multiplicative generalized cohomology theory and $E \rightarrow X$ a real vector bundle.
1. Is it true that, if $E$ is orientable with respect to h\*, then it is also orientable with respect to the singular cohomology with coefficients in $h^{0}(pt)$, for $(pt)$ a space with one point?
2. Is it true that... | https://mathoverflow.net/users/10758 | Orientation and generalized cohomology theories | Assuming $X$ is connected and the dimension $k$ of $\xi$ is positive, here is a proof of the yes answer to (1).
Consider the relative Atiyah-Hirzebruch spectral sequence
$$H^p(D\xi,S\xi;h^q(\mathrm{pt}))\Rightarrow h^\ast(D\xi,S\xi)$$
where $D\xi$ and $S\xi$ are the disk and sphere bundles in some Euclidean metri... | 10 | https://mathoverflow.net/users/8103 | 54116 | 33,805 |
https://mathoverflow.net/questions/54092 | 7 | In writing up a paper, we need references (and help) for the following
facts which are probably well-known. They concern morphisms of complex algebraic
varieties as continious maps in complex topology. (Here by 'complex topology' I mean
the topology induced by the metric on $\Bbb C$).
(i) Is every algebraic morphi... | https://mathoverflow.net/users/4745 | complex algebraic morphisms as topological maps: every morphism is a topological fibration on a Zariski dense open subset? | I have also wondered about question (i) in the past, and fortunately the answer is yes.
Here is a reference:
Verdier, Stratification de Whitney et theoreme de Bertini-Sard, Invent 1976, Cor 5.1
The result is probably also contained in Thom, Bull AMS 75 (1969), but it may
be harder to extract (at least it was for m... | 5 | https://mathoverflow.net/users/4144 | 54123 | 33,808 |
https://mathoverflow.net/questions/54093 | 8 | For a surjective morphism $f \colon X \to Z$ of smooth projective varieties, a line bundle $L$ on $X$ is $f$-nef if is nef on the fibers of $f$. More precisely, for every curve $C$ that is mapped to a point, the degree of $L$ restricted to $C$ should be larger than zero.
Now it is not true that an $f$-nef bundle bec... | https://mathoverflow.net/users/473 | When does f-nef imply nef (after twisting?) | Let $f=\alpha\circ g$ the Stein factorization of $f$ with $g:X\to W$ having connected fibers and $\alpha:W\to Z$ finite.
Let $x\in {\rm supp}\, D\subset X$ and $w=f(x)$ and let $C\subseteq X\_w=g^{-1}(w)$ such that $x\in C$. Since $-D\cdot C\geq 0$ this means that $C\subseteq {\rm supp}\, D$. Therefore, ${\rm supp}\,... | 10 | https://mathoverflow.net/users/10076 | 54125 | 33,809 |
https://mathoverflow.net/questions/53758 | 17 | Assume one is given a commutative square of spaces
$A \quad \to \quad C$
$
\downarrow \qquad \qquad \downarrow$
$B\quad \to \quad X$
which is a pushout and in which each map is a cofibration.
If $A \to B$ is $r$-connected and $A\to C$ is $s$-connected,
then the Blakers-Massey theorem says that the square is
$... | https://mathoverflow.net/users/8032 | Transversality in the proof of the Blakers-Massey Theorem. Is it necessary? | Take a look at the proof (attributed to Puppe) given in tom Dieck's new algebraic topology texbook (section 6.9). (I believe it also appears in tom Dieck, Kamps, Puppe (Lecture Notes in Mathematics 157).) This argument contains no obvious appeal to transversality.
| 9 | https://mathoverflow.net/users/437 | 54126 | 33,810 |
https://mathoverflow.net/questions/54071 | 5 | Let $G$ be a compact Lie group acting effectively on a compact, Hausdorff topological space $X$. I am looking for results of the type
*If $X$ is a ... and the action is ... then $\dim(X/G)\leq \dim(X)-\dim(G)$*.
Here $\dim$ denotes the [covering dimension](http://en.wikipedia.org/wiki/Lebesgue_covering_dimension).... | https://mathoverflow.net/users/8103 | Dimensions of orbit spaces | I'm loathe to answer my own question, but...
Theorem IV.3.8 of Bredon's book [*Introduction to compact transformation groups*](http://books.google.co.uk/books?id=qXEl_AIeSBUC&printsec=frontcover&dq=introduction+to+compact+transformation+groups&source=bl&ots=w_Rpcfjv_F&sig=C8MiAvvNNsAwgf7Ki3VUixSe3Zk&hl=en&ei=nrtJTevi... | 5 | https://mathoverflow.net/users/8103 | 54131 | 33,812 |
https://mathoverflow.net/questions/54122 | 51 | I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer examination it does not even seem well-defined. The formula is the one that I was given as the definition of the cross-product... | https://mathoverflow.net/users/7311 | Does this formula have a rigorous meaning, or is it merely formal? | But there is a commutative ring available, along the lines of what Mariano says. If $k$ is a field and $V$ is a vector space, then $k \oplus V$ is a commutative ring by the rule that a scalar times a scalar, or a scalar times a vector, or a vector times a scalar, are all what you think they are. The only missing part i... | 31 | https://mathoverflow.net/users/1450 | 54132 | 33,813 |
https://mathoverflow.net/questions/54130 | 2 | Given a random vector $(X\_1,X\_2)$. If $aX\_1 + bX\_2$ is Gaussian for all pairs $a,b$, then $(X\_1,X\_2)$ is jointly normal. More generally, is the following statement true?
If $aX\_1 + bX\_2$ has the same distribution as $aY\_1 + bY\_2$, for all $a,b$, then $(X\_1,X\_2)$ has the same distribution as $(Y\_1,Y\_2)$. I... | https://mathoverflow.net/users/4923 | when does inner product with fixed vectors determine joint distribution? | The first question has an affirmative answer. If two 2-dimensional distributions have the same (2-dim) characteristic functions they coincide. The characteristic function of the 2-dimensional distribution of (X,Y) is determined only by the distributions of aX+bY for all a and b.
The proof of that fact (Fourier inversi... | 5 | https://mathoverflow.net/users/6921 | 54133 | 33,814 |
https://mathoverflow.net/questions/54128 | 13 | There are many, many examples in mathematics of operations $s$ satisfying $ss = s$ (i.e., idempotent operations).
Not quite as common, but still numerous, are operations $s$ satisfying $sss = s$, specifically, Galois connections from a poset to itself; see my recent post [Abstract nonsense attribution](https://mathov... | https://mathoverflow.net/users/3621 | singly-generated monoids in mathematics | Think about the sheaves on some site as a full subcategory of presheaves: $Sh(C)\to PSh(C)$. This has a left adjoint, called sheafification. There are various ways to construct the sheafification, but one of them uses something called the plus-construction. For any presheaf $F$ it gives an associated presheaf $F^+$, wh... | 18 | https://mathoverflow.net/users/10862 | 54157 | 33,830 |
https://mathoverflow.net/questions/54144 | 3 | Let $\gamma(i)$ be a self avoiding walk (SAW) on a 2D lattice $L$ (a square lattice for example) starting at a predefined origin ( $\gamma(0)=(0,0)$ ) and having length $n:=\ell(\gamma)$. Furthermore, suppose $\gamma$ ends at a point $u$, so that $\gamma(n)=u$. Let $N(u)$ be the set of neighboring vertices of $u$ (that... | https://mathoverflow.net/users/934 | Self Avoiding Walk Pair Correlation | If the walks are sampled uniformly at random, then the probability of another neighbour of $u$ being visited is related to the "atmosphere" of a walk, and the connective constant $\mu$. The mean number of additional neighbours approaches $3-\mu \approx 0.361841469\cdots$ for SAWs on $Z^2$, while the probability of ther... | 8 | https://mathoverflow.net/users/6363 | 54158 | 33,831 |
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