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https://mathoverflow.net/questions/202337 | 5 | Throughout let $B$ be a stable C\*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C\*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any countably generated Hilbert $B$-module $X$ is singly generated, i.e. there exists a positive element $b\in B$ such that $X... | https://mathoverflow.net/users/16023 | When are countably generated Hilbert modules generated by c.p.c. order zero maps? | If $A=\mathbb C$ then the answer to both questions is yes.
If $A=M\_2(\mathbb C)$, then the modules $H=\overline{\phi(A)B}$ are those that have a direct sum decomposition $H\cong E\oplus E$ (where $E= \overline{\phi(e\_{1,1})B}$).
It is clear that not all modules need to have this property. The answer to the second q... | 4 | https://mathoverflow.net/users/13381 | 203132 | 97,892 |
https://mathoverflow.net/questions/203130 | 1 | The [heuristic justification](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Heuristic_justification) section of the [Wikipedia article about Goldbach's conjecture](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture) says that the argument that suggests that
*the number of twin primes below $x$ should be roug... | https://mathoverflow.net/users/13625 | Primes as uncorrelated random variables | As I understand your question, we have random variables $X\_n$, $n\in\mathbb N$ taking values in $\{0,1\}$.
We intuitively think of $X\_n=1$ as ``$n$ is prime'', but other than that this has little to do with primes.
Letting $p\_n:=\Pr(X\_n=1)$, we have that heuristically $p\_n\rightarrow 0$ at a certain rate, and ... | 3 | https://mathoverflow.net/users/4600 | 203135 | 97,894 |
https://mathoverflow.net/questions/203117 | 4 | Are there examples of polynomials $x\_1(t), x\_2(t) \in \mathbb{Q}[t]$ of equal degree at least one, with $\gcd(x\_1(t), x\_2(t)) = 1$, such that the sum $(x\_1(t))^4 + (x\_2(t))^4$ is divisible by the square of a polynomial $z(t)$ which is defined over $\mathbb{Q}$? If this question is too much to ask for, can we find... | https://mathoverflow.net/users/10898 | Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root | Here's an example: $x\_1(t)=3t^4 + 4t - 1$, $x\_2(t)=t^4 - 4t^3 - 3$. Then
$\begin{align\*}(x\_1&(t))^4 + (x\_2(t))^4=\\
&82t^{16} - 16t^{15} + 96t^{14} + 176t^{13} + 136t^{12} + 144t^{11} + 288t^{10} + 336t^9 +\\
&108t^8 + 336t^7 + 288t^6 + 144t^5 + 136t^4 + 176t^3 +96t^2 - 16t + 82=\\
&2\cdot(t^4 + 1)^2 \cdot (41t^... | 13 | https://mathoverflow.net/users/32216 | 203140 | 97,897 |
https://mathoverflow.net/questions/203138 | 7 | Apparently one identifies the configuration space in physics often with a manifold $M$. The tangent bundle $TM$ is then the space of all possible positions and velocities.
Furthermore, many sources seem to claim that $T^\*M$ can be regarded as the phase space, where $(q,p) \in T^\*M$ satisfies by definition that $p \... | https://mathoverflow.net/users/69763 | Momentum a cotangent vector | The Lagrangian is a function on the tangent bundle $L:TM\rightarrow\mathbb{R}$. Given a point $q\in M$ and a Lagrangian, we can define a function $L\_q:T\_qM\rightarrow \mathbb{R}$ using the simple formula $L\_q(v\_q)=L(v\_q)$, where $v\_q\in T\_qM$ is a tangent vector at $q\in M$. Notice that $L\_q$ is a mapping betwe... | 9 | https://mathoverflow.net/users/27121 | 203159 | 97,904 |
https://mathoverflow.net/questions/203145 | 2 | This question is related to the MO questions [What is the difference between Grothendieck groups K\_0(X) vs K^0(X) on schemes?](https://mathoverflow.net/questions/22120/what-is-the-difference-between-grothendieck-groups-k-0x-vs-k0x-on-schemes) and [Does a fully faithful functor between triangulated categories induce em... | https://mathoverflow.net/users/24965 | Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$? | No, this is not always a monomorphism. For the underlying reduced scheme $X\_{\text{red}}$ of $X$, the pushforward homomorphism
$$
K\_0(X\_{\text{red}})\to K\_0(X)
$$
is an isomorphism (via devissage). If you read Manin's "Lectures on the K-functor", you will see that the natural map
$$
\text{Pic}(X) \to K^0(X)
$$
... | 4 | https://mathoverflow.net/users/13265 | 203164 | 97,906 |
https://mathoverflow.net/questions/202820 | 7 | Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find such material by searching on the web. This doesn't seem to be a difficult problem, but maybe there is a neat and memorable... | https://mathoverflow.net/users/39762 | Integrating a barycentric monomial over a simplex | Not for the first time in my life, and almost certainly not the last, I have to eat humble pie. I thought that Igor had missed the point in his comment. In fact, it was I who missed the point. My answer has the advantage that it uses only very elementary methods. It needed a strong nudge and explanations from Igor befo... | 8 | https://mathoverflow.net/users/39762 | 203166 | 97,907 |
https://mathoverflow.net/questions/203161 | 1 | Is there a connected topological space that is maximal compact, but not $T\_2$? (A space $(X,\tau)$ is said to be *maximal compact* if for any topology $\tau'$ on $X$ with $\tau'\supseteq \tau$ and $\tau'\neq \tau$ we have that $(X,\tau')$ is not compact.)
| https://mathoverflow.net/users/8628 | Connected, maximal compact, but not $T_2$ | Let $X$ be any compactly generated connected Hausdorff space that is not locally compact and let $Y=X\cup\{\infty\}$ be its one-point compactification. Then $Y$ is compact, connected, and not Hausdorff. To show $Y$ is maximal compact, we must show that every compact subset $K\subseteq Y$ is closed in $Y$. If $K\subsete... | 4 | https://mathoverflow.net/users/75 | 203168 | 97,909 |
https://mathoverflow.net/questions/203147 | 15 | The Thompson group Th of order $90745943887872000$ is one of the sporadic simple groups occurring in the classification of finite simple groups.
Its maximal subgroups are known (see <http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Th/>) and they are all remarkably small, which has the consequence that any permutation r... | https://mathoverflow.net/users/1492 | Explicit permutation representation of the Thompson sporadic simple group? | The following Magma code worked in less than an hour. It is using the idea suggested by Dima Pasechnik working with the group ${\rm Th} < {\rm GL}(248,2)$, acting on an orbit of a subspace of dimension $2$ fixed by the maximal subgroup $^3D\_4(2):3$. It used about 43GB. The group is coming from the ATLAS database, whic... | 14 | https://mathoverflow.net/users/35840 | 203180 | 97,914 |
https://mathoverflow.net/questions/203189 | 7 | I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets (<http://en.wikipedia.org/wiki/Complete_Heyting_algebra>), but $C$ is not completely distributive (<http://en.wikipedia.org/... | https://mathoverflow.net/users/70653 | Counterexample on completely distributive lattices | A complete Boolean algebra is both a frame and a co-frame. It is completely distributive if and only if it is atomic (see for example [here](http://thue.stanford.edu/bool.html)), so all you need is a non-atomic complete Boolean algebra. The lattice of regular open sets (an open set is *regular* if it is equal to the in... | 13 | https://mathoverflow.net/users/2926 | 203193 | 97,918 |
https://mathoverflow.net/questions/203172 | 8 | Let $M$ be a real analytic manifold. In the book "Sheaves on Manifolds" by Kashiwara and Schapira it is claimed on p. 127 (without reference or proof) that
1. the Poincare lemma holds for the de Rham complex of real analytic differential forms,
2. the cohomology of the sheaves of real analytic analytic differential $... | https://mathoverflow.net/users/13302 | Acyclicity of the sheaf of real analytic differential forms | Regarding question 2 :This is answered in the paper of Henri Cartan Bulletin SMF vol 85 yr 1957 pages 77-99 .It is essentially as outlined by user74230 .Question 1:If you look at the proof of Poincare lemma in Narasimhan's book Analysis on Real and Complex manifolds pages 128 to 129 it is clear that the proof works for... | 7 | https://mathoverflow.net/users/4696 | 203196 | 97,919 |
https://mathoverflow.net/questions/203118 | 14 | Let $X$ and $Z$ be smooth manifolds and $\phi: X \to Z$ a smooth map so that the differential $D \phi$ is everywhere of rank $d$. Is there necessarily a $d$-fold $Y$ so that $\phi$ factors as a submersion $X \to Y$ followed by an immersion $Y \to Z$?
I have no real motivation, it just seemed like a natural question. ... | https://mathoverflow.net/users/297 | Factoring constant rank maps into a submersion and an immersion | A smooth factorization does not exist in all cases.
Let $X$ be $\mathbb{R}^2$ minus the closed ray $R = \{ (x,0)\ |\ x\ge 0\}$, and define $\phi:X\to\mathbb{R}^2 = Z$ by the rule
$$
\phi(x,y) = \begin{cases}(x,0), & \text{when $x\le 0$,}\\
\bigl(x,\phantom{-}e^{-1/x^2}\bigr), & \text{when $x> 0$ and $y>0$.}\\
\bigl(... | 19 | https://mathoverflow.net/users/13972 | 203201 | 97,921 |
https://mathoverflow.net/questions/202943 | 7 | The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded.
Here I ask whether another possible generalization (for which I could not yet find a name starting with L) holds or not.
Suppose that for some events $\mathbf A$ we have a dependency graph and an as... | https://mathoverflow.net/users/955 | Is there a Degenerate Dependency Local Lemma? | Here is my intuition that it may not be possible.
I am guessing that as in the case of the original LLL, such an inequality would in turn imply a simpler inequality of the following form:
"If the dependency graph is $d$-degenerate and every event has probability at most $p$ and $4pd<1$,
then we can avoid all event... | 5 | https://mathoverflow.net/users/10926 | 203214 | 97,927 |
https://mathoverflow.net/questions/203215 | 10 | Is there a difference between Euler systems and Kolyvagin systems - or do they refer to the same thing? For example there is the Heegner point Euler system, but you don't really see a Heegner point Kolyvagin system.
Also, Rubin-Mazur prove that the space of Kolyvagin system is free of rank one (under certain conditi... | https://mathoverflow.net/users/70666 | What's the difference between Euler systems and Kolyvagin systems? | Euler systems and Kolyvagin systems are closely related but quite different beasts nevertheless.
According to their respective definitions, Euler systems are systems of classes $\{c(n)\in H^1(G\_{\mathbb Q(\zeta\_n)},V)\}\_n$ for $V$ a $p$-adic $G\_{\mathbb Q}$-representation verifying certain properties, especially ... | 10 | https://mathoverflow.net/users/2284 | 203218 | 97,928 |
https://mathoverflow.net/questions/137290 | 14 | Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a [polycyclic-by-finite group](http://en.wikipedia.org/wiki/Polycyclic_group).
Denote the (Banach) space (in fact an algebra under convolution) $$\ell^1(G)=\le... | https://mathoverflow.net/users/9305 | $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $\ell^1(G)$ is a flat (right) $\mathbb{Z}G$-module? | The general answer is no for my question.
For example, for $G=\mathbb{Z}^2\rtimes\mathbb{Z}$ with exponential growth rate, $\ell^1(G)$ is not flat over $\mathbb{Z}G$, the proof is based on rather elementary calculation. But it is too long to be present here..
The point is to find $f\in\mathbb{Z}G\cap\ell^1(G)^{\ti... | 4 | https://mathoverflow.net/users/9305 | 203222 | 97,929 |
https://mathoverflow.net/questions/203049 | 8 | Consider the standard embedded $n$-cross polytope $P\_n$ with vertices $\pm e\_i \in \mathbb R^n$. Let us consider decompositions of this polytope into $2^{n-1}$ simplices, such that these simplices have only vertices that are also vertices of $P\_n$. How many different such decompositions exist? (Note that such a deco... | https://mathoverflow.net/users/14233 | Decomposition of a cross-polytope into simplices | The cross-polytope has $2n$ vertices in $n$ antipodal pairs. A simplex with $n+1$ points from these must contain exactly one antipodal pair of points, since if it contains more than one, it has a square face and $0$ volume. So, the simplex is the union of two adjacent cones over facets, or the intersection of the cross... | 9 | https://mathoverflow.net/users/2954 | 203226 | 97,931 |
https://mathoverflow.net/questions/203228 | 4 | Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $L$ with respect to $\langle\;,\; \rangle$. We let $L^\*$ denote the dual lattice of $L$ with respect to $\langle\;,\rang... | https://mathoverflow.net/users/11765 | Fourier coefficients of real analytic functions on an n-dimension torus | By the compactness of the torus, there is a uniform radius of convergence $r>0$ working for every point. You can extend $f$ to complex variables and use Cauchy's formula to find $|\partial^k f|\le Ck!(r/2)^{-k}$, with the constant $C$ only depending on $\sup\_{|\Im z|\le r} |f|$. Then you get the same sort of bound for... | 5 | https://mathoverflow.net/users/37103 | 203232 | 97,933 |
https://mathoverflow.net/questions/203184 | 10 | By work of Deligne and others (I am following Deligne-Milne's notes which I just began to read: <http://www.jmilne.org/math/xnotes/tc.pdf>) we know that a given affine group scheme G can be recovered from their category of representations Rep(G) (thought of as a neutral Tannakian category). If G-->G' is a morphism Prop... | https://mathoverflow.net/users/70647 | Exact sequences of groups and Tannakian formalism | About your main question I suggest looking at Appendix A in
On Nori's Fundamental Group Scheme
Hélène Esnault, Phùng Hô Hai, Xiaotao Sun
Geometry and dynamics of groups and spaces, 377–398, Progr. Math., 265, Birkhäuser, Basel, 2008.
<http://arxiv.org/abs/math/0605645>
<http://link.springer.com/chapter/10.1007... | 7 | https://mathoverflow.net/users/11682 | 203235 | 97,935 |
https://mathoverflow.net/questions/203127 | 5 | If I have a parallelogram $P$ symmetric around the origin, and a vector $v$, such that $(P+v)\cap (P-v)$ is not empty, is there a simple way to obtain the parallelogram $Q\subset (P+v) \cup (P-v)$, symmetric around the origin, with the biggest area?
It seems that if $v$ is very small it is better to take the sides of... | https://mathoverflow.net/users/39359 | Biggest parallelogram inside the union of two translated parallelograms | We may assume that the two parallelograms are squares (e.g. applying a suitable linear transformation, or, what is the same, choosing an Euclidean structure in the plane, that induce that measure, and for which the parallelograms are squares). At least, this simplifies the notation and reduces the number of data.
The... | 3 | https://mathoverflow.net/users/6101 | 203239 | 97,937 |
https://mathoverflow.net/questions/203249 | 0 | Let us call a space $(X,\tau)$ *totally separated (ts)* if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq \tau$ the space $(X,\sigma)$ no longer has this property we call $(X,\tau)$ *minimal ts*.
If $(X,\tau)$ is ts, ... | https://mathoverflow.net/users/8628 | Minimal totally separated spaces | First, note that a minimal totally separated space is the same thing as a Stone space. Clearly Stone spaces are minimal totally separated (any coarser topology cannot even be Hausdorff); conversely suppose $X$ is totally separated and not Stone. We may assume the topology on $X$ is generated by its clopen sets (otherwi... | 5 | https://mathoverflow.net/users/75 | 203252 | 97,940 |
https://mathoverflow.net/questions/203254 | 7 | I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which one should I start with? and can you recommend books on either one?
| https://mathoverflow.net/users/70684 | Studying topology: which first, algebraic or differential? | Run, don't walk, to Milnor's "Topology From The Differentiable Viewpoint."
| 10 | https://mathoverflow.net/users/1231 | 203257 | 97,942 |
https://mathoverflow.net/questions/203265 | 9 | In his paper *QFT and Jones Polynomials*, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated surgeries on knots." (page 383).
Is there an analogous statement/theorem for 4-manifolds? Such as obtaining other 4-manifol... | https://mathoverflow.net/users/44768 | Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$ | If an $n$-dimensional smooth manifold $X'$ is obtained from $X$ by doing some surgeries, then there is an $(n+1)$-dimensional smooth cobordism $W$ from $X$ to $X'$; vice-versa, any handle decomposition of such a cobordism induces a sequence of surgeries.
Hence, the equivalence relation of "being obtained from one ano... | 18 | https://mathoverflow.net/users/13119 | 203268 | 97,945 |
https://mathoverflow.net/questions/203182 | 34 | This puzzle is taken from the book *Mathematical puzzles: a connoisseur's collection* by *P. Winkler*.
>
> Two sheriffs in neighboring towns are on the track of a killer, in a
> case involving eight suspects. By virtue of independent, reliable
> detective work, each has narrowed his list to only two. Now they ar... | https://mathoverflow.net/users/5712 | "The Two Sheriffs" puzzle | Here's a solution for the case of seven suspects that uses the Fano plane. Let the seven points of the Fano plane represent the seven suspects. Alice and Bob both reveal the name of the suspect completing a line with the two suspects on their list. There are now two cases to consider:
1. Alice and Bob did not name su... | 46 | https://mathoverflow.net/users/20186 | 203270 | 97,946 |
https://mathoverflow.net/questions/203273 | 1 | Given $g\in\mathcal{C}^1(\bar\Delta)$, and $z\in\Delta$, how can i prove that the 2-form
$$
d\omega=\frac{\partial\_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta}
$$
is integrable in $z$?
At MSE no one could help me, I hope to be more lucky here.
If you need more detail look at
<https://math.stackexchange... | https://mathoverflow.net/users/70148 | Integrability at $z$ of the 2-form $ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $ | Griffiths & Harris (pp. 2-3) put it thus:
"Setting $\zeta - z=re^{i\theta}$,
$$
d\zeta\wedge d\bar\zeta = -2i dx\wedge dy = -2i rdr\wedge d\theta
$$
so
$$
\left|\frac{\partial g(\zeta)}{\partial\bar\zeta}\frac{d\zeta\wedge d\bar\zeta}{\zeta - z}\right|
= 2\left|\frac{\partial g}{\partial\bar\zeta}dr\wedge d\theta\rig... | 1 | https://mathoverflow.net/users/19276 | 203278 | 97,947 |
https://mathoverflow.net/questions/203285 | 2 | For finite groups $L \leq K$ we define $d(L,K)$ to be the least $n \in \mathbb{N}$ for which there exist $a\_1, \dots, a\_n \in K$ such that $\langle L, a\_1 \dots, a\_n \rangle = K$.
Is there some $m \in \mathbb{N}$ for which the following claim holds:
Let $G$ be a finite group and let $H \leq G$ be a subgroup wi... | https://mathoverflow.net/users/38889 | Generating finite groups using subgroups | I don't think so.
Let $K$ be any group, and let $e\neq t\in C\_2$ act on $K\times K$ by exchanging coordinates. Let $G=(K\times K)\rtimes \langle t\rangle$, and let $H$ be generated by the elements $(k,1)$ with $k\in K$. Then $\langle H,t\rangle = G$, so $d(H,G) = 1$.
Now let $G\_0 = K\times K$. Then $d(H,G\_0)=d(... | 6 | https://mathoverflow.net/users/3959 | 203287 | 97,951 |
https://mathoverflow.net/questions/203255 | 10 | Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, but a priori one does not know if the first moments are equal.
(It is also known that all negative moments exist, but ... | https://mathoverflow.net/users/70686 | A moment problem | The argument from the paper Carlo linked to (which, by the way, is essentially a classical result of Cramer's) can be adapted to your situation.
Consider the moment generating function $f(z) = Ee^{z\ln X}$ of the random variable $\ln X$. By your assumption, this is well defined for $0\le \textrm{Re}\, z<1/2$. Moreove... | 8 | https://mathoverflow.net/users/48839 | 203292 | 97,953 |
https://mathoverflow.net/questions/203313 | 6 | Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.
>
> Suppose there exists some point $x \in X$ whose stabiliser $G\_x$ is trivial. Does there exist an open subset $U \subset X$ such that the stabiliser $G\_u$ is trivial for al... | https://mathoverflow.net/users/5101 | Stabilisers of group actions | Under some of your extra hypotheses, namely if $G$ is reductive and the orbit $Gx$ is closed, the answer is yes. This follows easily from Luna's slice theorem; see for instance [these lectures](http://webusers.imj-prg.fr/~jean-marc.drezet/papers/Wykno.pdf), Proposition 5.7.
| 7 | https://mathoverflow.net/users/40297 | 203316 | 97,964 |
https://mathoverflow.net/questions/202990 | 3 | In this [question](https://mathoverflow.net/questions/202956/a-conjecture-about-parallelizable-generalized-spheres), we have discussed how the following bundle:
$E\_{d} = TS^{d}\oplus \Lambda^2 T^{\ast}S^{d}$
is always trivial, where $S^{d}$ is the $d$-dimensional standard sphere. Now, let us take $d=6$. The six-sp... | https://mathoverflow.net/users/66688 | Parallelizable nearly-Kahler manifolds | The answer is 'no'. Already, it is not true for $M = \mathrm{Sp}(2)/\bigl(\mathrm{SU}(2)\times\mathrm{U}(1)\bigr)$, which is known to be diffeomorphic to $\mathbb{CP}^3$.
To see this, note that the first Pontrjagin class of $T\mathbb{CP}^3$ satisfies
$$
p\_1(T\mathbb{CP}^3) = 8u^2\in H^4(\mathbb{CP}^3,\mathbb{R}),
$$... | 6 | https://mathoverflow.net/users/13972 | 203327 | 97,967 |
https://mathoverflow.net/questions/203290 | 7 | Say that the $n$-th prime $p\_n$ is *isolated to degree $k$*
(my notation) if
the prime gap to either side is larger than $\log p\_n$ to the $k$-th power:
\begin{eqnarray\*}
p\_n - p\_{n-1} & > & (\log p\_n)^k \;,\\
p\_{n+1} - p\_n & > & (\log p\_n)^k \;,
\end{eqnarray\*}
where $\log$ is the natural log.
*Examples*.... | https://mathoverflow.net/users/6094 | Primes isolated by large gaps to either side | Currently it is not even known, for any $k>1$, if $p\_{n+1}-p\_n>(\log p\_n)^k$ holds infinitely often. The best known result in this direction is due to Ford, Green, Konyagin, Maynard, Tao, see [here](http://arxiv.org/abs/1412.5029).
On the other hand, as Jeremy Rouse explained in two comments, it is expected that t... | 4 | https://mathoverflow.net/users/11919 | 203332 | 97,968 |
https://mathoverflow.net/questions/203330 | 13 | Let $V$ be a finite dimensional vector space.
Let us call an automorphism $T:V\rightarrow V$ **admissible** if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry.
We know $T$ is admissible if and only if there exists a basis for $V$ such that the representing matrix of $T$ w.r.t to this b... | https://mathoverflow.net/users/46290 | Is there a global obstruction for a diffeomorphism to be an isometry? | Yes, there are global obstructions. Consider $M= S^1 \times \mathbb R$ and $\phi$ acting by an irrational rotation on $S^1$ and by multiplication by $2$ on $\mathbb R$. There are no finite orbits, but also no invariant metrics: The function that takes a point $P$ on $S^1$ to the length of a vector pointing along $\math... | 14 | https://mathoverflow.net/users/18060 | 203335 | 97,970 |
https://mathoverflow.net/questions/203338 | 0 | Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E\_0\subset E\_1\cdots\subset E\_s=E$ be the Harder-Narasimhan filtration. Suppose $E\_1$ is $\mu$-stable and rank $E\_1$ is 2. Then the paper says that $E$ sits in the following short exact sequence :
$0\longrightarrow E\_1\longr... | https://mathoverflow.net/users/70211 | Why does this vector bundle on the surface sit in this exact sequence? | Any torsion free sheaf on a smooth surface can be written in the form $N \otimes I\_\xi$, in particular this holds for $E/E\_1$. Explicitly, one can write $N = \det(E)\otimes \det(E\_1)^{-1}$, and also one can figure out the length of $\xi$ from Chern classes of $E\_1$ and $E$. The actual subscheme $\xi$ is more diffic... | 1 | https://mathoverflow.net/users/4428 | 203339 | 97,972 |
https://mathoverflow.net/questions/203202 | 14 | I was reading a little about knots (in a popular math book that wasn't very good) and the book put forth several knot invariants like the Alexander and Jones polynomials. But these are not complete invariants. This made me wonder "what are some complete knot invariants?" and the obvious next question "what is the descr... | https://mathoverflow.net/users/6342 | Descriptive Complexity of Knot Equivalence | A perfect timing for this question, since I just uploaded a paper on this topic to arXive (see below). Let us specify the definitions. A knot is a homeomorphic image of the circle in $\mathbb{R}^3$. Two knots are equivalent, if there exists a homeomorphism of the ambient space onto itself taking one knot to the other. ... | 12 | https://mathoverflow.net/users/56461 | 203346 | 97,975 |
https://mathoverflow.net/questions/203348 | 16 | Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of
$$w\_p = \prod\_{a \in (\mathbb F\_p^{\times})^2}(1 + \zeta^a) + \prod\_{b \in \mathbb F\_p^{\times} \backslash (\mathbb F\_p^{\times})^2}(1 + \zeta^b). $$
(The term $w\_p$ is related to the num... | https://mathoverflow.net/users/14233 | Evaluating a remarkable term for primes p = 5 (mod. 8) | According to Theorem 5.1 of
<http://math.mit.edu/~rstan/pubs/pubfiles/35.pdf>, the number is
$$ \frac 1p\left[ 2^{p-1}+\frac{p-1}{2}(\epsilon^{4h} +
\epsilon^{-4h})\right], $$
where $h$ is the class number of $\mathbb{Q}(\sqrt{p})$ and
$\epsilon>1$ the fundamental unit of $\mathbb{Q}(\sqrt{p})$.
**Addendum.** I sho... | 27 | https://mathoverflow.net/users/2807 | 203350 | 97,976 |
https://mathoverflow.net/questions/203297 | 4 | Suppose you are given a power series $P=\sum\_{i=0}^\infty{a\_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form
$$ \frac{1}{\prod\_{i=1}^k{(1-t^{\alpha\_i})}}.$$ My motivation comes from looking at Hilbert series of noetherian freely generated subrings of polynomial ri... | https://mathoverflow.net/users/41283 | Raising coefficients of a power series to some power | It's a general fact about any rational power series
$$\sum\_{n=0}^\infty a\_n t^n = {p(t) \over \prod\_{i=1}^r (1-\gamma\_i t)^{d\_i}}$$
(where $p(t)$ is some polynomial of degree less than $\sum\_{i=1}^r d\_i$ and the $\gamma\_i$ are distinct) that
$$a\_n = \sum\_{i=1}^r p\_i(n)\gamma\_i^n$$
for some polynomia... | 3 | https://mathoverflow.net/users/3106 | 203351 | 97,977 |
https://mathoverflow.net/questions/203322 | 1 | Let $\Omega\subseteq\Bbb C^2$ be open bounded (and connected), $f:\Omega\to\Bbb C$ separately holomorphic (i.e. $f$ is holomorphic in each variable when the other is fixed).
Hartogs theorem is not allowed, so we can't say $f$ is holomorphic on $\Omega$.
Now I think that in this case $f$ is continous in $\bar\Omega$... | https://mathoverflow.net/users/70148 | If $f$ is separately holomorphic on $\Omega$ then $f\in\mathcal{C}^0(\bar\Omega)\Leftrightarrow f\in L^1(\Omega)$ | This will not work if $L^1$ refers to area measure: the function $f(w,z)=1/z$ on (let's say) $|w|,|z|<1$, $z\not= 0$ is a counterexample.
| 2 | https://mathoverflow.net/users/48839 | 203359 | 97,979 |
https://mathoverflow.net/questions/203367 | 13 | Let $S$ be a scheme and let $X := \mathrm{Spec}(\mathscr{O}\_S[t, t^{-1}])$ be the underlying $S$-scheme of the $S$-group scheme $(\mathbb{G}\_m)\_S$. Is there only one structure of a commutative $S$-group scheme on $X$ for which $t = 1$ is the identity section?
| https://mathoverflow.net/users/63877 | Is there a unique commutative group structure on $\mathbb{G}_m$? | Yes if $S$ is reduced, and no otherwise. The case of a field is classical (ultimately because $k[t,1/t]^{\times} = k^{\times}\cdot t^{\mathbf{Z}}$ for fields $k$), and I assume you are familiar with that. So in general if $S$ is reduced then by passing to the case of affine $S$ and then noetherian $S$ (by the usual lim... | 27 | https://mathoverflow.net/users/70739 | 203375 | 97,984 |
https://mathoverflow.net/questions/201388 | 3 | I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.
---
*Pointwise Lagrange interpolation*
Given a function $f \in C^0([-1,1])$ and a grid of $n+1$ nodes $X = (x\_i)\_{i = 0}^n$ on $[-1,1]$ we ... | https://mathoverflow.net/users/68687 | Relation between Chebyshev Interpolation and Expansion | There is an excellent explanation of this in Chapter 4 of L. N. Trefethen's [Approximation Theory and Approximation Practice](https://people.maths.ox.ac.uk/trefethen/ATAP/) (henceforth ATAP; the first 6 chapters are [available for free online](https://people.maths.ox.ac.uk/trefethen/ATAP/ATAPfirst6chapters.pdf)). I wil... | 4 | https://mathoverflow.net/users/20507 | 203379 | 97,986 |
https://mathoverflow.net/questions/203331 | 5 | Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}\_p$ and let $E$ be another such extension, such that all the $\mathbb{Q}\_p$ embeddings $K \to \bar{\mathbb{Q}}\_p$ are contained in $E$.
Let $V$ be an $n$-dimensional vector space over $E$ which carries a continuous action of $G\_K = Gal(\bar... | https://mathoverflow.net/users/60691 | Hodge-Tate weights of induced representation | It's simpler (and more general) to merely assume that your representations are de Rham. If $W$ is a de Rham representation of $G\_L$, then $D = D\_{dR}(W)$ is an $L$-vector space. If $K$ is a subfield of $L$ then $D\_{dR}(Ind\_L^K W)$ is $D$, now seen as a $K$-vector space via restriction of scalars. The rest should th... | 5 | https://mathoverflow.net/users/5743 | 203380 | 97,987 |
https://mathoverflow.net/questions/203062 | 3 | Recall that a strongly minimal theory $T$ has the Definable Multiplicity Property (DMP) if for all natural $k$, $m$ and $\varphi(\bar{x},\bar{b})$ of rank $k$, multiplicity $m$, there exists a formula $\theta ∈ tp(\bar{b})$ such that for all $\bar{b^{'}}\models\theta$, $rk(\varphi(\bar{x},\bar{b^{'}}))=k$ and $mult(ϕ(\... | https://mathoverflow.net/users/38966 | Strongly minimal set with DMP | The answer is no, there is a counter example by Hasson and Hrushovski, constructed using Fraisse limit. It is described in their article "DMP in strongly minimal sets", J. Symbolic Logic
Volume 72, Issue 3 (2007), 1019-1030.
| 4 | https://mathoverflow.net/users/2234 | 203381 | 97,988 |
https://mathoverflow.net/questions/202672 | 4 | I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by
$$ \|f\|=\|\hat f\|\_{L^p} $$
for $p\in [1,\infty]$. I find it handy to give it a name when discussing the success/failure of Young's inequality on the Fourier transform, among other things.
| https://mathoverflow.net/users/37103 | Is there a name for this space? | In the literature you can sometimes see them called **Fourier-Lebesgue** spaces, with notation $\mathcal{F} L^p(\mathbb{R}^n)$, consisting of the set of all tempered distributions whose norm (as you wrote) is finite.
See, e.g., <http://arxiv.org/abs/0804.1730> and <http://arxiv.org/abs/0801.1444>
| 5 | https://mathoverflow.net/users/3948 | 203383 | 97,990 |
https://mathoverflow.net/questions/203295 | 7 | I recently needed to know which circles $S$ in a maximal torus $T^6$ of the compact exceptional group $E\_6$ yield one-dimensional subspaces $\mathfrak s$ of the Lie algebra $\mathfrak t^6$ that are reflected by some element of the Weyl group $W\_{E\_6}$.
I found that such ($-1$)-eigenspaces were precisely those con... | https://mathoverflow.net/users/5792 | Is this characterization of (-1)-eigenspaces of the Weyl group of $E_6$ known? | EDIT II: Sorry to bump this again but the answer to this question can be phrased entirely in terms of finite Coxeter groups and doesn’t depend at all on the fact that we’re dealing with $\textsf{E}\_6$, so it seemed best to write the answer that way.
Let $V$ be a finite dimensional real euclidean vector space and let... | 4 | https://mathoverflow.net/users/22846 | 203386 | 97,992 |
https://mathoverflow.net/questions/203384 | 1 | Let $R$ be a closed integral domain with its fraction field $F$. Let $K$ be a finite separable extension field of $F$, and let $A$ be the integral closure of $R$ in $K$.
It is well known that the trace map $Tr: K \to F$ is non-trivial and hence is surjective because of separable extension. If we restrict $Tr$ to $A$,... | https://mathoverflow.net/users/56989 | Surjectivity of trace map | No, the trace map need not be surjective on the level of rings. This is one of the difficulties of "wild ramification". For instance, let $k$ be a field of characteristic $p$, let $R$ be $k[x]$, and let $A$ be the $R$-algebra,
$$ A = R[y]/\langle y^p +xy-1 \rangle = k[x,y]/\langle y^p + xy-1 \rangle.$$
Using the Ja... | 2 | https://mathoverflow.net/users/13265 | 203388 | 97,993 |
https://mathoverflow.net/questions/203387 | 10 | Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative monoid" structure Banach and Hilbert space-like space?
Thanks for your help.
| https://mathoverflow.net/users/68372 | What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space? | Let me expand my comments in an short answer.
A *(left) semimodule $M$ over a semiring $R$* is a commutative monoid $(M, \, +)$ together with a multiplication map $R \times M \to M$, denoted by $(r, \, m) \to rm$ and called *scalar multiplication*, which satisfy all axioms of a unitary ring except the axiom demanding... | 18 | https://mathoverflow.net/users/7460 | 203389 | 97,994 |
https://mathoverflow.net/questions/203397 | 3 | I am looking for a version of the Ballot Theorem for general step distributions. Specifically, let $X\_1,X\_2,\ldots$ be i.i.d. real random variables with some distribution. Let $S\_n = S\_1 + \cdots + S\_n$. Let
$$ p\_n = \mathbb{P}[S\_1 > 0,\ldots,S\_n > 0].$$
For $X\_n$ supported on $\{+1,-1\}$ this can be calcula... | https://mathoverflow.net/users/23661 | General ballot theorem | You might want to check out the following nice [survey](http://cgm.cs.mcgill.ca/~reedbook/papers/BallotTheorems.pdf) of Ballot theorems by Addario-Berry and Reed. For example, Theorem 3 (due to Takács) deals with the case that each $X\_i$ is integer valued with mean $\mu$ and maximum value $1$. As you only care about a... | 3 | https://mathoverflow.net/users/2233 | 203401 | 97,999 |
https://mathoverflow.net/questions/203407 | 19 | **Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a reasonable moduli space, is it finite or infinite dimensional?**
Remark. For $n=1$ there exist exactly two complex struct... | https://mathoverflow.net/users/16183 | Classification of complex structures on $\mathbb{R}^{2n}$ | There exist *infinitely many* inequivalent complex structures in $\mathbb{R}^{2n}$ for all $n \geq 2$.
See for instance the paper
K. Diederich, N. Sibony: *[Strange complex structures on Euclidean space](https://eudml.org/doc/152188)*, Journal für die reine und angewandte Mathematik **311-312**, page 397-407 (1979... | 20 | https://mathoverflow.net/users/7460 | 203408 | 98,002 |
https://mathoverflow.net/questions/203409 | 3 | **Definitions:** Let $\omega^{<\omega}$ be the set of all finite sequences of natural numbers. For $u, v \in \omega^{<\omega}$, let $u \prec v$ denote that $u$ is a prefix of $v$. We call a subset $T \subseteq \omega^{<\omega}$ a *tree*, if $v \in T$ and $u \prec v$ imply $u \in T$. We say that $T$ contains an infinite... | https://mathoverflow.net/users/15002 | Illfounded trees as "retract" of all trees | Suppose, toward a contradiction, that $R$ is a function of the sort you asked about. Then, for any tree $T$, we have the equivalence "$T$ has an infinite path $\iff$ every infinite path through $R(T)$ is a path through $T$." The implication from left to right is the second of your requirements for $R$. The converse is ... | 3 | https://mathoverflow.net/users/6794 | 203413 | 98,004 |
https://mathoverflow.net/questions/203391 | 2 | Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C\_{q^{\infty}}$ onto $A$ such that $A$ does not contain proper uncountable $C\_{p^{\infty}}$-invariant subgroups?
| https://mathoverflow.net/users/66046 | Uncountable cardinals and Prufer $p$-groups | The answer is no:
>
> For all primes $p,q$, for every uncountable elementary abelian $p$-group, for every action of the Prüfer $q$-group $C\_{q^\infty}\simeq\mathbf{Z}[1/q]/\mathbf{Z}$ on $A$ by group automorphisms, there exists a proper uncountable $C\_{q^\infty}$-invariant subgroup in $A$.
>
>
>
There are 2 ... | 5 | https://mathoverflow.net/users/14094 | 203416 | 98,005 |
https://mathoverflow.net/questions/203309 | 3 | Let $(g\_i)\_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda\_i)\_{i=1,...,d}$ non-random s.t. $\max\_i(\lambda\_i)=1$ and $\lambda\_i>0, \forall i$.
I am looking for the expectation of the Mahalanobis norm $E[\sqrt{\sum\_{i=1}^d \lambda\_i g\_i^2}]$.
I know it in the special case when all $\la... | https://mathoverflow.net/users/61472 | Expectation of Mahalanobis norm | A greater and more general lower bound holds:
$$(\*)\qquad E f\Big(\sum\_{i=1}^d \lambda\_i g\_i^2\Big) \ge E f(X\_\lambda),$$
where $\lambda:=\lambda\_1+\dots+\lambda\_d$, $X\_\lambda$ has the $\chi^2$ distribution with $\lambda$ degrees of freedom (that is, the $\text{Gamma}(\lambda/2,2)$ distribution), and $f$ is ... | 3 | https://mathoverflow.net/users/36721 | 203418 | 98,006 |
https://mathoverflow.net/questions/185140 | 1 | Let $q$ denote a quadratic form over a field $k$.
The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$.
Let $k = \mathbb{Q}\_p$ for any prime $p$ and set
$L = k(t\_1,..,t\_n)$.
It is known that $u(k)=4$ and newer results by David B. Leep state that... | https://mathoverflow.net/users/51251 | Cohomological dimension of transcendental p-adic extensions | I think the answer to your question can be found in Serre's "Galois cohomology" book, Section II.4.2, where a more general result is proved for transcendental field extensions.
| 2 | https://mathoverflow.net/users/11599 | 203421 | 98,007 |
https://mathoverflow.net/questions/203411 | 3 | (This is a repost of a question from math.SE, <https://math.stackexchange.com/questions/1240966/existence-of-state-on-a-c-algebra-satisfying-tauab-ab>)
Let $a,b$ be elements of a unital C\*-algebra $A$ with $0\leq a,b\leq 1$. Is it the case there is a state $\tau$ on $A$ such that $|\tau(ab)|=\|ab\|$? I'm particular... | https://mathoverflow.net/users/16107 | Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$ | I think there are $2\times 2$ counterexamples. Take $a = \pmatrix{1&0\cr 0&0}$ and
$b = \pmatrix{.5&.5\cr .5&.5}$, so that $c = ab = \pmatrix{.5&.5\cr 0&0}$. If there were a state $\tau$ satisfying $\tau(c) = \lambda$ for some $|\lambda| = \|c\|$ then there would be a face of the state space which does this, so there i... | 6 | https://mathoverflow.net/users/23141 | 203425 | 98,008 |
https://mathoverflow.net/questions/203393 | 8 | I am currently reading something about nonholomorphic Eisenstein series $E\_\mathfrak{a}(z,1/2+it)$ for $\Gamma\_0(q)$, where $\mathfrak{a}$ is a cusp (cf. Iwaniec, H. *Spectral Methods of Automorphic Forms*). For Maass cusp forms there is Atkin-Lehner theory, but I don't know whether there are similar results for $E\_... | https://mathoverflow.net/users/70741 | Atkin-Lehner theory for nonholomorphic Eisenstein series | I don't believe such a theory exists anywhere in the literature. To the best of my knowledge, there are a couple of results that deal closely with what you are asking.
There exists a theory of newforms for Eisenstein series of the form
\[E\_{\chi\_1,\chi\_2}(z,k,\varepsilon) = \sum\_{n = 0}^{\infty} a\_n e(nz),\]
whe... | 10 | https://mathoverflow.net/users/3803 | 203426 | 98,009 |
https://mathoverflow.net/questions/203423 | 3 | Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K\_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K\_{\nu}^{unr} $. Let $E $ be an elliptic curve defined over $K $, and let $E\_{p^n} $ denote its $p^n $-divisors, where $p $ is a (rational) prime that is not a multiple of $\n... | https://mathoverflow.net/users/70751 | A question on the cohomology of elliptic curves over local fields | By the Kummer sequence, your kernel is isomorphic to $E(K\_\nu)/p^nE(K\_\nu)$ via the connecting homomorphism. If you also assume that your elliptic curve has good reduction ath $\nu$, then the cocycle you get via the connecting homomorphism will be unramified, which I think is what you want (plus use inflation-restric... | 4 | https://mathoverflow.net/users/11926 | 203429 | 98,010 |
https://mathoverflow.net/questions/182747 | 5 | There are many results on the cohomology of the Hilbert scheme of points of a surface.
Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also
there are results on the ring structure of the cohomology. In Manfred Lehn and Christoph
Sorger's paper "The cup product of the Hilbert sc... | https://mathoverflow.net/users/59172 | On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces | I did do this recently.
Have a look here:
<http://arxiv.org/abs/1410.8398>
and here for the source code:
<https://github.com/s--kapfer/HilbK3>
| 2 | https://mathoverflow.net/users/62593 | 203431 | 98,011 |
https://mathoverflow.net/questions/203448 | 3 | For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is not weakly equivalent to $X \times \Omega X$.
| https://mathoverflow.net/users/14233 | Example s.t. the unbased loop-space is not $\Omega X \times X$ | Let $G$ be a discrete group. Let $P$ denote the groupoid whose objects are the elements of $G$, and whose morphisms from $a$ to $b$ are the elements $g\in G$ such that $gag^{-1}=b$. It is then a standard fact that $\Omega'BG=BP$. Less naturally but more concretely, if we choose a set $C$ of representatives for the conj... | 6 | https://mathoverflow.net/users/10366 | 203450 | 98,017 |
https://mathoverflow.net/questions/203449 | 9 | I just read this question [link](https://mathoverflow.net/questions/2615/least-number-of-charts-to-describe-a-given-manifold) and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this special situation there is an e... | https://mathoverflow.net/users/69763 | Charts needed for an atlas | You are looking for the smallest number of open contractible sets needed to cover $M.$ This is so well studied, that it has a name: \*the Lyusternik-Shnirelman category of $M$." For references, you can look at [the nice paper by Gomez-Larranaga, Heil, and Gonzalez-Acuna](http://www.math.fsu.edu/~aluffi/archive/paper260... | 9 | https://mathoverflow.net/users/11142 | 203452 | 98,018 |
https://mathoverflow.net/questions/203453 | 10 | For $x\_i \in \mathbb{Z}$, let $\{x\_i\}$ be a *fundamental* solution to the equations:
$$
\sum\_{i= 1}^N x\_i = \sum\_{i=1}^N x\_i^3 = 0
$$
if $x \in \{x\_i\} \Rightarrow -x \notin \{x\_i\}$.
For instance, a fundamental solution with $N=7$ is given by
$$
x\_1 = 4, \quad x\_2 = x\_3 = x\_4 = -3, \quad x\_5 = x\_6 =... | https://mathoverflow.net/users/nan | Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$ | We can get $N=6$ from $1+5+5=2+3+6$, $1^3+5^3+5^3=2^3+3^3+6^3$.
We can get $N=5$ from $2+4+10=7+9$, $2^3+4^3+10^3=7^3+9^3$.
| 12 | https://mathoverflow.net/users/3684 | 203454 | 98,019 |
https://mathoverflow.net/questions/203445 | 21 | Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T\_X$ (not only canonical bundle $K\_X$), is it true that $X$ is an abelian variety?
I think the holomorphic tangent bundle of a Hopf surface will not be trivial, according to the comment below..
| https://mathoverflow.net/users/nan | Must an algebraic variety with trivial tangent bundle be an abelian variety? | More generally, in the complex case the following result holds.
>
> **Theorem.** Let $X$ be a compact Kähler manifold which is complex parallelisable, i.e. such that
> $T\_X$ is holomorphically trivial. Then $X$ is a complex torus. In particular, if $X$ is algebraic then $X$ is an abelian variety.
>
>
>
For ... | 28 | https://mathoverflow.net/users/7460 | 203479 | 98,028 |
https://mathoverflow.net/questions/203477 | 7 | Is there a smooth solution to minimize this:
$$
\int\_0^1{x \over {1+k^2f'(x)^2}}dx, f(0)=1, f(1)=0, f'(x)\leq 0, k^2>0.
$$
I could "solve" it using a numeric approximation (my algorithm converged so apparently there is a valid local minimum in the function space) but would love to see an analytic solution (algebraic o... | https://mathoverflow.net/users/70779 | Functional minimization problem | Actually this is one of the oldest problems in the calculus of variations. It's named "the Newton problem" after Sir Isaac Newton, who studied it in 1685. It arises from the determination of the optimal profile for the motion of bodies (projectiles, ships, etc), that is, the profile giving the minimal aerodynamic or hy... | 11 | https://mathoverflow.net/users/6101 | 203484 | 98,030 |
https://mathoverflow.net/questions/203478 | 1 | Let $(X,\tau)$ be a topological space.
We say that $(X,\tau)$ is *zero-dimensional with respect to the Lebesgue covering dimension (zd1)* if every open cover of the space has a refinement which is a cover of the space by open sets such that any point in the space is contained in exactly one open set of this refinemen... | https://mathoverflow.net/users/8628 | Two notions of zero-dimensionality for topological spaces | Take the Sierpiński space $X=\{0,1\}$ with $\tau=\{\{\},\{0\},\{0,1\}\}$.
| 3 | https://mathoverflow.net/users/3075 | 203485 | 98,031 |
https://mathoverflow.net/questions/203488 | 5 | This problem came up in a PDE where I used separation of variables to formally get a solution. Now I need to know whether that formal solution is sensible.
Let $a\_k >0$ be an increasing sequence of real numbers. Let $f\_k$ be real numbers.
Let $R \in (0,\infty)$ and $y \in (0,\infty)$. Let
$$v(y) = \sum\_{k=1}^\i... | https://mathoverflow.net/users/70786 | Does this infinite sum arising from separation of variables converge? | For fixed $R > 0$, $B\_1(k,R) \sim 1$ as $k$ goes to infinity, while $B\_2(k,R) \sim e^{- 2 a\_k R}$. The bound on $B\_1$ combined with the summability of the $f\_k$ takes care of one half of your sum.
The other half has a general term of order $f\_k e^{-2a\_kR + a\_ky}$. If the $f\_k$ are only generic summable reals... | 4 | https://mathoverflow.net/users/62629 | 203489 | 98,033 |
https://mathoverflow.net/questions/203481 | 2 | Let $π\colon Y = \mathrm{Proj}\_B \mathcal{A} \rightarrow B$ be a morphism constructed from a coherent graded sheaf of $\mathcal{O}\_B$-algebras $\mathcal{A} = \bigoplus\_k \mathcal{A}$.
I am looking for (minimal) hypotheses for the natural morphism
$$
\mathcal{A\_k} \rightarrow \mathcal π\_\*O\_Y(k)
$$
to be surjec... | https://mathoverflow.net/users/56926 | Relative Proj and generation of sections | If the base is affine Noetherian, this is discussed in [this section of the Stacks project](http://stacks.math.columbia.edu/tag/01YR). See especially the last two lemmas with graded module $M = A$. Also, I want to mention that I do not know what you mean by a "coherent sheaf of $\mathcal{O}\_B$-algebras".
| 1 | https://mathoverflow.net/users/68366 | 203490 | 98,034 |
https://mathoverflow.net/questions/203505 | 21 | Let $P(x)$ be a non-constant polynomial with real coefficients.
Can [natural density](http://en.wikipedia.org/wiki/Natural_density) of
$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$
be positive?
| https://mathoverflow.net/users/38805 | A Polynomial With Positive Prime Density | No. There are two cases. Firstly, suppose that one of the non-constant coefficients of $P$ is irrational. Then, by the Weyl equidistribution theorem, $\lfloor P(n) \rfloor$ is equidistributed mod $W$ for any modulus $W$, which already limits the natural density of the prime-producing $n$ to be at most $\phi(W)/W$ for a... | 24 | https://mathoverflow.net/users/766 | 203521 | 98,044 |
https://mathoverflow.net/questions/203519 | 4 | My question is somehow related to the one here [First Collision Time for k Random Walkers on a Torus](https://mathoverflow.net/questions/169297/first-collision-time-for-k-random-walkers-on-a-torus) but, unfortunately, the answer does not cover my concern.
My problem is: consider $n$ walkers on the cycle $\mathbb{Z}/k... | https://mathoverflow.net/users/59239 | First collision time of $n$ random walkers on a cycle | Suppose the initial distribution of walkers has each independently choosing a site with equal probabilities (note that multiply-occupied sites are allowed).
This distribution is invariant under the process. The probability of a collision at any step is the probability that the site the moving walker enters is occupied,... | 0 | https://mathoverflow.net/users/13650 | 203523 | 98,046 |
https://mathoverflow.net/questions/203499 | 20 | This question was originaly posted in the stackexchange <https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures> but since it only got a comment I decided to post it here. I don't know if this is allow, please let me know if it's not. anyway...
I've been reading about optimal... | https://mathoverflow.net/users/70345 | A measure on the space of probability measures | von Renesse and Sturm have constructed a family "natural" measures $\mu$ making $(\mathcal{P}([0,1]),d\_{W\_2},\mu\_\beta)$ into a metric measure space: <http://www.ams.org/mathscinet-getitem?mr=2537551>. They argue that their measure can be *formally* thought of as
$$
\tag{\*} \mu\_\beta = C\_\beta^{-1} e^{-\beta Ent... | 19 | https://mathoverflow.net/users/1540 | 203532 | 98,049 |
https://mathoverflow.net/questions/203398 | 4 | It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the projectiviziation of the light cone. In other words, if $Q(x)=Q(y)=0$ for non-zero vectors $x,y \in \mathbb{R}^n$, then... | https://mathoverflow.net/users/3635 | Orbits of the maximal compact subgroup on the light cone for $p$-adic groups | Let $(V,(x,y)\mapsto x.y)$ be a bilinear space. Let $L$ be a lattice on $V$, and let $G$ be its stabilizer in $O(V)$.
To an isotropic line $\ell$ in $V$ one can associate the ideal $I(\ell):=(\ell\cap L).L$, and of course, two lines $\ell\_0$ and $\ell\_1$ belong to a same orbit under $G$ only if $I(\ell\_0)=I(\ell\... | 0 | https://mathoverflow.net/users/39552 | 203537 | 98,050 |
https://mathoverflow.net/questions/203509 | 9 |
>
> **Notations** For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written in the form
> \begin{equation}
> f(s)=\sum\_{n=1}^{\infty}{\frac{a\_n}{n^s}} \nonumber
> \end{equation}
> where ... | https://mathoverflow.net/users/66686 | Is this theorem on $L$-functions known? | Closely related problems have been extensively studied; in particular much stronger versions of the corollaries are already known. Here are some references: Fujii was the first to show that a positive proportion of the zeros of two different Dirichlet $L$-functions are
different. A stronger version of this, namely of ... | 10 | https://mathoverflow.net/users/38624 | 203545 | 98,052 |
https://mathoverflow.net/questions/203538 | 2 | Let $X$ be a subset of $\mathbb{R}^d$, let $\|\cdot \|\_p$ be a norm with $1\leq p\leq\infty$, and let $f:\mathbb{R}^d\to\mathbb{R}$ be a function. I'm trying to find examples of $X$, $p$, and $f$ for which it is computationally tractable to minimize the function $$\|x\|\_p-f(x)$$ over $x\in X$, and in particular, to t... | https://mathoverflow.net/users/70190 | Functions that are easy to compare to a norm | If $f$ is homogeneous, you can just try to minimize it on the unit ball (which is a Lagrange multiplier problem, which does not mean it's easy), and see if any of your critical values are smaller than $1\dots$
| 1 | https://mathoverflow.net/users/11142 | 203551 | 98,054 |
https://mathoverflow.net/questions/203550 | 6 | Let $\Bbb{F}\_q$ be a finite field. Choose a non-square $\delta \in \Bbb{F}\_q^\*$
and form the quadratic extension $\Bbb{F}\_q\big( \sqrt{\delta} \, \big)$. For
an element $z \in \Bbb{F}\_q\big( \sqrt{\delta} \, \big)$ let $\text{N}(z) := zz^q$ be its norm. Select a non-trivial multiplicative character
$\chi: \Bbb{F}... | https://mathoverflow.net/users/70119 | Finite field "contour" sum | What you call finite field contour sums are more usually called exponential sums. Like complex analytic contour integrals, they do not always (or even usually) have a nice exact formula. Instead, sometimes they have a completely satisfactory exact formula, sometimes they have an expression in terms of special functions... | 14 | https://mathoverflow.net/users/18060 | 203556 | 98,057 |
https://mathoverflow.net/questions/203565 | 6 | In a synthetic (Pappian) projective plane, one can define a conic in various clever ways not referring to coordinates. For instance, if $f$ is a projectivity from the pencil of lines through a point $A$ to the pencil of lines through another point $B$, then the locus of intersections $m \cap f(m)$ is a conic (which is ... | https://mathoverflow.net/users/49 | Synthetic projective definition of cubic curves | It is a "classical" fact that any nonsingular plane cubic curve $C$ can be projectively generated by means of a pencil of lines and a pencil of conics.
The starting point of the construction is the observation that the lines defined by any $g\_2^1$ on $C$ all pass through the same point $p$ of $C$, that following Syl... | 10 | https://mathoverflow.net/users/7460 | 203568 | 98,059 |
https://mathoverflow.net/questions/203576 | 8 | [Vopěnka's principle](http://ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle) is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle then no locally presentable category contains a full subcate... | https://mathoverflow.net/users/184 | Explicit counter example to Vopěnka's principle in the constructible universe? | This is a counterexample to Vopenka's principle phrased slightly differently: as "in any proper class of first-order structures, one elementarily embeds into the other."
Working in $V=L$, I claim that $\{L\_{\kappa^+}: \kappa\in Card\}$ is a counterexample to Vopenka's principle.
Suppose $\kappa<\lambda$ are cardin... | 8 | https://mathoverflow.net/users/8133 | 203582 | 98,064 |
https://mathoverflow.net/questions/203590 | 1 | Throughout this post, let $(P,\leq)$ be a poset. The *interval topology* $\tau\_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
Now we define the *order convergence topology*... | https://mathoverflow.net/users/nan | Interval topology and order convergence topology | The answer is Yes, and it suffices to show that for $p\in P$ we have $(P\setminus \downarrow p) \in \tau\_o(P)$ for all $p\in P$. (A similar argument then also shows that $(P\setminus \downarrow p) \in \tau\_o(P)$, which proves that a subbasis of $\tau\_i(P)$ is contained in $\tau\_o(P)$.)
Let $x\in P\setminus \downa... | 1 | https://mathoverflow.net/users/8628 | 203592 | 98,066 |
https://mathoverflow.net/questions/203591 | 0 | Consider the associative algebra A with generators $T\_i$ and rule $T\_i\*T\_j=\Sigma\_kC^{ij}\_k\*T\_k$. Even if it makes no sense for a fusion ring (my momentary pet :-) to change basis it is still possible to linear transform the basis (via $T\_j=M\_{ij}E\_i$) into diagonal form: $E\_i\*E\_j=\delta\_{ij}\*E\_i$. (Or... | https://mathoverflow.net/users/11504 | "Diagonalizing" an associative algebra | I misread your question. My comment alluded to the question whether you can have a basis such that $E\_i \* E\_j = $ *some* single $E\_k$, (as opposed to a linear combination of all of them) but what you ask is much stronger: does every commutative associative algebra have a basis of orthogonal idempotents. This implie... | 3 | https://mathoverflow.net/users/41139 | 203596 | 98,068 |
https://mathoverflow.net/questions/203601 | 7 | Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots then, Is it true that $f$ has to be a zero polynomial? Here $m<k$ is a positive integer. This statement is true for $m=1$ but... | https://mathoverflow.net/users/62241 | Common roots of polynomial and its derivative | Assume $a,b,c \in \mathbb{R}$ solve
$$2(a^3+b^3+c^3)-3(a^2b+ab^2+b^2c+bc^2+a^2c+ac^2)+12abc=0,$$
e.g. $(a,b,c)=(-1,1,3)$. Then
$$
\begin{eqnarray}
f(x)&:=&(x-a)(x-b)(x-c)(3x^2-2(a+b+c)x+3(ab+bc+ca)-2(a^2+b^2+c^2))\\
&=&3x^5-5(a+b+c)x^4+10(ab+bc+ca)x^3\\
&&+(2(a^3+b^3+c^3)-3(a^2b+ab^2+\dots)-18abc)x^2+\dots
\end{eqnarr... | 6 | https://mathoverflow.net/users/35593 | 203610 | 98,074 |
https://mathoverflow.net/questions/203569 | 2 | A space $(X,\tau)$ is said to have *Lebesgue covering dimension $\leq n$* (for some $n\in\mathbb{N}$) if every open covering $\cal U$ has a refinement $\cal V$ such that for every $x\in X$ the set ${\cal V}\_x := \{V\in \mathcal{V}: x\in V\}$ has at less than $n$ elements. (Note that $n$ is "globally fixed" for all ope... | https://mathoverflow.net/users/8628 | Lebesgue covering dimension for locales | It is easy to see that a space $X$ has Lebesgue covering dimension less than or equal to $n$ if and only if whenever $\mathcal{U}$ is an open covering, there is some refinement $\mathcal{V}$ so that whenever $O\_{0},...,O\_{n+1}\in\mathcal{V}$ are distinct we have $O\_{0}\cap...\cap O\_{n+1}=\emptyset$. This notion can... | 2 | https://mathoverflow.net/users/22277 | 203618 | 98,076 |
https://mathoverflow.net/questions/203572 | 4 | Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
This question is strongly linked to
[is the space of all borel measures on $\mathbb{R}^n$ isomorphic to the tensor ...](https://mathoverflow.net/questions/202323/is-the-space-of-all-borel-measure... | https://mathoverflow.net/users/25256 | Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$ | In [stereotype theory](http://en.wikipedia.org/wiki/Stereotype_space) there is an isomorphism of stereotype spaces (or, what is the same here, an isomorphism of locally convex spaces)
$$
{\mathcal C}^\star(X)\circledast {\mathcal C}^\star(Y)\cong{\mathcal C}^\star(X\times Y),
$$
where $X$ and $Y$ are arbitrary paracomp... | 1 | https://mathoverflow.net/users/18943 | 203626 | 98,078 |
https://mathoverflow.net/questions/203461 | 2 | Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad\_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the Killing form. Finding the Lie algebra of $H$ would be desirable.
| https://mathoverflow.net/users/41654 | Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$ | This is really an extended comment, but, because it's too long to put into a comment box and because it may help answer some of the OP's questions, I'm putting it here.
If one endows $\mathrm{SU}(n)$ with its usual bi-invariant measure $\mathrm{d}\mu$ normalized to have total volume $1$ (aka Haar measure), then one k... | 2 | https://mathoverflow.net/users/13972 | 203630 | 98,079 |
https://mathoverflow.net/questions/203624 | 3 | Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. How about $\mathscr{F}'$ if the other two sheaves are flasque? Can we prove $\mathscr{F}'$ flasque or end the question by... | https://mathoverflow.net/users/44250 | A question on flasque sheaf | For a naturally-arising example, take $X$ to be a smooth variety and consider
$0 \to \mathcal{O}\_X^\times \to \mathcal{R}\_X^\times \to \mathcal{D}\textit{iv}\_X \to 0$
where $\mathcal{O}\_X$ is the structure sheaf, $\mathcal{R}\_X$ is the sheaf of rational functions, and $\mathcal{D}\textit{iv}\_X$ is (by definit... | 2 | https://mathoverflow.net/users/3753 | 203638 | 98,084 |
https://mathoverflow.net/questions/203633 | 0 | Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X \times\_R Q$ the base change of $X$. Is $Y$ a regular scheme? If not true in general, is there any additional assumption on ... | https://mathoverflow.net/users/58203 | Base change of regular schemes | No. Take $R=\mathbb{Z}\_p$ and $X$ to be the affine scheme defined by $xy=p$. This is regular, but its base change to $\mathbb{Z}\_p[\sqrt{p}]$ is not.
If $X$ is smooth over $R$, then $Y$ is smooth over $Q$ (and therefore regular in many useful situations). This is of course a much stronger assumption.
| 2 | https://mathoverflow.net/users/3753 | 203639 | 98,085 |
https://mathoverflow.net/questions/203643 | 2 | Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."):
"Assume $(x\_n)$ converges weakly to $x$. Then there exists a sequence $(y\_n)$ made up of convex combinations of the $x\_n$'s that converges strongly to $x$.''
The Lemma says that "there exists a sequence...''.
Is it true that $\textbf{every}$ ... | https://mathoverflow.net/users/70760 | A question involving Mazur's Lemma | If $x\_i= u\_{floor(log(i)+1)}$ where $\{u\_i| i=1,2,\ldots\}$ is an orthonormal basis for a Hilbert Space, then $(x\_i)$ converges weakly to $0$ and $y\_n = 1/n \sum\_{i=1}^n x\_i$ does not converge strongly to anything.
In particular, if $n = floor( e^k )$ for an integer $k\geq1$, then $||y\_n - u\_k|| < 2/3$ henc... | 6 | https://mathoverflow.net/users/70858 | 203646 | 98,088 |
https://mathoverflow.net/questions/203628 | 3 | First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^\*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of $K$.
In a paper I read, the author uses the fact that there's a continuous function $g$ s.t. $\int gdm>sup\_{\mu \in K... | https://mathoverflow.net/users/70853 | How to show that there's a continuous function separating convex sets of Radon measures? | Since the weak topology is Hausdorff, there is an open set containing $m$ and disjoint from $K$. So we can find $f\_1,...,f\_n$ and $\varepsilon$ such that the open set
$$\{\mu \mid \int f\_i dm -\varepsilon < \int f\_i d\mu < \int f\_i dm +\varepsilon \quad \forall \ i \}$$
is disjoint from $K$. Now project everything... | 2 | https://mathoverflow.net/users/6129 | 203650 | 98,091 |
https://mathoverflow.net/questions/203636 | 5 | Suppose $\mathcal{A}$ is a commutative Banach algebra (over $\mathbb{R}$) or commutative Banach \*-algebra (over $\mathbb{C}$). Is there always a measurable space $(\Omega,\mathcal{F})$ such that there is a bijective correspondence between states (positive linear functionals of norm 1) on $\mathcal{A}$ and probability ... | https://mathoverflow.net/users/68564 | Can states on commutative Banach algebras be understood as probability measures? | 1. The Gelfand space and transform you mention are defined for any commutative **Banach algebra** $A$, as $A^\dagger=\{$nonzero linear functionals $\chi:A\to\mathbf C:\chi(ab)=\chi(a)\chi(b)\}$ and $\hat a(\chi)=\chi(a)$.
2. On the other hand, the definition of a *state* $m$ on $A$ asks that $m$ be positive ($m(a^\*a)\... | 8 | https://mathoverflow.net/users/19276 | 203651 | 98,092 |
https://mathoverflow.net/questions/203654 | 1 | Draw $n$ numbers, denoted by $a\_1, a\_2, \ldots, a\_n$, from set $[n]$, that is, for each $i$, $a\_i$ is a uniformly random number from $[n]$.
Let $A = \{a\_1, a\_2, \ldots, a\_n\}$. Then
$$
\mathbb{E}[|A|] = n - n (\frac{n-1}{n})^n \approx n (1 - 1/e).
$$
How to prove a concentration bound on $|A|$? For example, pr... | https://mathoverflow.net/users/26659 | Balls from bin with replacement, distinct elements, concentration inequality | We have that
$$
P(|A|\le cn) \le \binom{n}{cn} c^n
$$
(first choose a subset of cardinality $cn$ and then insist that all numbers come from this set). By Stirling's formula, this bound is asymptotically
$$
\sim \frac{1}{\sqrt{n}} \left( \frac{c}{1-c} \right)^{(1-c)n} ,
$$
which is of the requested type as long as $c<1/... | 2 | https://mathoverflow.net/users/48839 | 203656 | 98,093 |
https://mathoverflow.net/questions/203422 | 6 | Good afternoon everyone !
I have the following question of Riemannian geometry :
Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth Riemannian metric on $M$ with sectional curvature pinched between $-1- \epsilon$ and $-1$ $\}$ where $\epsilon$ is an arbitrar... | https://mathoverflow.net/users/25511 | Negatively curved metrics minimizing the length of a homotopy class of simple closed curves | We have $0<\inf\_{g\in \mathcal{T}} L\_g(\gamma) \leq \sup\_{g\in \mathcal{T}} L\_g(\gamma) <\infty$. In fact, there should be a universal bound on the ratio
$\sup\_{g\in \mathcal{T}} L\_g(\gamma)/ \inf\_{g\in \mathcal{T}} L\_g(\gamma)$ for all $\gamma \in \pi\_1 M$.
This follows from a [theorem of Belegradek](http:/... | 2 | https://mathoverflow.net/users/1345 | 203665 | 98,095 |
https://mathoverflow.net/questions/203673 | 6 | Let $X$ be a locally compact Hausdorff space. Does there exist a locally finite open covering consisting of relatively compact sets?
| https://mathoverflow.net/users/nan | Does every locally compact Hausdorff space admit a locally finite open covering by relatively compact sets? | Not necessarily. The ordinal space $\omega\_1 = [ 0 , \omega\_1 )$ provides a counterexample.
To see that there is no locally finite cover by relatively compact sets, note that every compact subset — and therefore every relatively compact subset — is bounded. So if $\mathcal{A}$ is a cover by relatively compact sets... | 6 | https://mathoverflow.net/users/13653 | 203678 | 98,098 |
https://mathoverflow.net/questions/203251 | 2 | Let us call a space $(X,\tau)$ [*totally separated*](http://en.wikipedia.org/wiki/Connected_space#Disconnected_spaces) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq \tau$ the space $(X,\sigma)$ no longer has this pr... | https://mathoverflow.net/users/8628 | Are all minimal totally separated spaces compact? | First, note that a minimal totally separated space is the same thing as a Stone space. Clearly Stone spaces are minimal totally separated (any coarser topology cannot even be Hausdorff); conversely suppose $X$ is totally separated and not Stone. We may assume the topology on $X$ is generated by its clopen sets (otherwi... | 1 | https://mathoverflow.net/users/nan | 203684 | 98,100 |
https://mathoverflow.net/questions/203602 | 23 | Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig)
Edit: You may assume knowledge of representation theory of finite groups (as in Serre), algebraic groups and étale cohomology.
| https://mathoverflow.net/users/nan | learning Deligne-Lusztig theory | You have given no indication as to your background, so the following imagines you don’t know anything. I have purposely left interesting things out as this is designed to get you from 0 to DL theory. Essentially this is what I would do if I could have my time over again, precisely in this order.
EDIT: Assuming the ba... | 43 | https://mathoverflow.net/users/22846 | 203691 | 98,103 |
https://mathoverflow.net/questions/203688 | 1 | I'm condusion on a statement in this page [comma object](http://ncatlab.org/nlab/show/comma+object) in $n$lab. It states:
>
> any strict comma object is a comma object, but the converse is not in general true.
>
>
>
My confusion is: the strict comma object trivially satisfies the $1$-dimensional univerality, b... | https://mathoverflow.net/users/43795 | Strict comma objects implies comma objects | Your understanding of the definition is incorrect, but that is probably because the cited nLab page is misleading. Let me spell it out a little bit more accurately:
>
> Let $\mathfrak{K}$ be a 2-category and let $f : A \to C$ and $g : B \to C$ be morphisms in $\mathfrak{K}$. A **strict comma object** $(f \downarrow... | 2 | https://mathoverflow.net/users/11640 | 203692 | 98,104 |
https://mathoverflow.net/questions/203701 | 9 | Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack?
In many cases I know, the moduli stack is smooth (e.g., complete intersections, rigid Calabi-Yau threefolds, Calabi-Yau's with $\tau=1$).... | https://mathoverflow.net/users/70881 | Singularities of the moduli stack of Calabi-Yau threefolds | Yes, Calabi-Yau manifolds have unobstructed deformations. This is due to Tian and Todorov; there is a nice algebraic proof in a paper by Kawamata, J. Algebraic Geom. 1 (1992), no. 2, 183–190.
| 12 | https://mathoverflow.net/users/40297 | 203705 | 98,107 |
https://mathoverflow.net/questions/203672 | 12 | It is known ([Golod and Shafarevich](http://en.wikipedia.org/wiki/Golod%E2%80%93Shafarevich_theorem)) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of unity (in particular, when $\zeta^p=1$ where $p$ is a p... | https://mathoverflow.net/users/nan | Class field towers | Let $\ell$ be an odd prime and $m$ an integer such that
$$|\{p|m,\ p\equiv1\operatorname{mod} \ell\}|\geq8.$$
Then Y.Furuta proved that $\mathbb Q(\zeta\_m)$ admits an infinite unramified $\ell$-class field tower (*Nagoya Math. Journal*,1972). In fact, I.Shparlinski proved using this result that $\mathbb Q(\zeta\_m)$ ... | 11 | https://mathoverflow.net/users/2284 | 203708 | 98,108 |
https://mathoverflow.net/questions/203709 | 1 | Given any triple of positive integers $(g',g,d)$ with $2g'-2\geq d(2g-2)$.
Does there always exist curves $C\_{g'},C\_g$ of genus $g',g$ with a degree $d$ morphism $f\colon C\_{g'}\to C\_g$?
If we fix $C\_g$ a curve of genus $g$, can we always find a a curve $C\_{g'}$ with genus $g'$ and a degree $d$ morphism $f\c... | https://mathoverflow.net/users/nan | Construction of curves and morphisms | Yes. Let me assume for simplicity $g\geq 2$ — the cases $g=0,1$ can be treated in the same way with slight modifications. Put $r=2g'-2-d(2g-2)$, and choose $r+1$ distinct points $p;p\_1,\ldots ,p\_r$ on $C\_g$. The group $\pi:=\pi \_1(C\_g\smallsetminus \{p\_1,\ldots ,p\_r\},p )$ is generated by $2g+r$ elements $a\_1,\... | 3 | https://mathoverflow.net/users/40297 | 203710 | 98,109 |
https://mathoverflow.net/questions/203670 | 1 | My question is the following : suppose you have a doubly-connected open set $\Omega \subset \mathbb{C}$, that is a domain bounded by 2 non-intersecting circles $C\_1$ (the interior) and $C\_2$ (the exterior).
This is well known (an exercise in Ahlfors's book) that there exists a conformal mapping $\varphi$ from $\Ome... | https://mathoverflow.net/users/69533 | Extension of conformal map and annulus | As Robert writes, the answer is positive for *round* circles, which follows from the fact that Möbius transformations map circles to circles (and you can map any circle contained in the unit disc to a circle with its centre at 0).
As Neil mentions, the answer is negative for arbitrary curves. Indeed, this is trivial ... | 3 | https://mathoverflow.net/users/3651 | 203711 | 98,110 |
https://mathoverflow.net/questions/195569 | 2 | Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the Hilbert-Chow morphism $S^{[n]} \rightarrow \operatorname{Sym}^n S$. Let $L$ be a divisor on $S$ and $\tilde L$ the correspo... | https://mathoverflow.net/users/19088 | Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points | Here is the paper to look at.
<http://arxiv.org/abs/math/9904095>
| 1 | https://mathoverflow.net/users/19088 | 203715 | 98,112 |
https://mathoverflow.net/questions/203719 | 0 | Let $C$ be a quasi-projective curve, $C\_i$ for $i=1,...,r$ are the irreducible components of $C$. Assume that $C\_i$ is non-singular and $F\_i$ locally free sheaf on $C\_i$ of the same rank for all $i$. Can we glue the sheaves $F\_i$ i.e., does there exists a locally free sheaf $F$ on $C$ such that $F|\_{C\_i} \cong F... | https://mathoverflow.net/users/43198 | Gluing locally free sheaves on curves | By induction one can assume that $k = 2$. Let $Z = C\_1 \cap C\_2$ be the scheme-theoretic intersection. Choose an isomorphism of $\varphi:F\_{1|Z} \to F\_{2|Z}$ (it exists since both are free of the same rank) and define $F$ from the exact sequence
$$
0 \to F \to F\_1 \oplus F\_2 \to F\_{1|Z} \to 0,
$$
where the secon... | 3 | https://mathoverflow.net/users/4428 | 203724 | 98,114 |
https://mathoverflow.net/questions/203712 | 1 | There is a theorem by Jean Taylor that says that an almost minimal set in $\mathbb{R}^3$ can be locally parametrize by the only three possible minimal cones in $\mathbb{R}^3$, the plane, an $Y$ times a line and all the faces you can make from the center of a tetrahedron and its vertices.
This question has already bee... | https://mathoverflow.net/users/30500 | Singularities in minimal surfaces | On one hand, the answer to the question
>
> Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of have the Y or the tetrahedron singularity?. It is easy to see in experiments that in real soap bubbles this singula... | 6 | https://mathoverflow.net/users/1540 | 203727 | 98,115 |
https://mathoverflow.net/questions/203734 | 5 | I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie **inside** an $n$-dimensional ellipse.
That is, given an $n$-dimensional ellipsoid with a center at $c \in \mathbb{R}^n$: $$E(A,c,r) := \Big\{x \in \mathbb{R}^n ~\Big| ~(x-c)^T A (x-c) \leq r\Big\}... | https://mathoverflow.net/users/49673 | Bounding the number of lattice points inside an $n$-dimensional ellipsoid | Well, it depends on what you want to know. For homothetic images $t E$ of a fixed ellipsoid $E,$ the quantity is asymptotic to $t^n vol(E),$ where the error term is $O(t^{n-1}),$ with better bounds provable. But maybe you want to know something else?!
**EDIT** For non-asymptotic bounds, see this [paper of Bentkus and... | 6 | https://mathoverflow.net/users/11142 | 203736 | 98,116 |
https://mathoverflow.net/questions/203731 | 6 | $\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have a smooth function $f$ defined on an open neighborhood $U$ of $0$ in $\bR^n$. To a smooth Riemannian metric $g$ defined ... | https://mathoverflow.net/users/20302 | An unusual metric reconstruction problem | If $V$ is a vector field such that $Vf=0$ at every point, then the 2-tensor
$$
h^{ij}=g^{ij}+V^iV^j
$$
is also an inverse metric and satisfies $\nabla^gf=\nabla^hf$.
In a coordinate chart this amounts to finding a vector field orthogonal (in the Euclidean sense on the chart) pointwise orthogonal to the gradient of $f$.... | 9 | https://mathoverflow.net/users/55893 | 203739 | 98,119 |
https://mathoverflow.net/questions/203735 | 3 | Let $D$ be an irreducible hermitian symmetric domain. Then there exists a variation of Hodge structures $(h\_s)\_{s\in D}$ on a vector space $V$ satisfying specific conditions which depend on $D$ such that the space $S=\{h\_s\}$ is isomorphic to $D$. Let $X$ be a smooth projective variety and let $\pi\colon X\to D$ be ... | https://mathoverflow.net/users/66288 | Variation of Hodge structures associated to a hermitian symmetric domain | You might want to look at the book "Mumford-Tate Groups and Domains: Their Geometry and Arithmetic", by Mark Green, Phillip Griffiths, and Matt Kerr. You should also look at the recent work of Colleen Robles. These might have answers to some of your questions.
| 2 | https://mathoverflow.net/users/13972 | 203740 | 98,120 |
https://mathoverflow.net/questions/203733 | 6 | I am reading the article *D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991)* ([click](https://math.dartmouth.edu/~carlp/PDF/paper82.pdf)).
In equation (27) the authors, apparently, used the following upper bound for a sum over the prime numbers:
$$(\star)\quad\sum\_{p... | https://mathoverflow.net/users/nan | Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$ | The left hand side of $(\star)$ is at least
$$ \sum\_{p\leq x}p^{-1}=\log\log x +O(1), $$
hence a necessary condition for the truth of $(\star)$ is that
$$ \frac{x^{\varepsilon}}{\varepsilon\log x} \gg \log\log x.$$
This condition is also sufficient, in the light of the following bound that I prove below:
$$ (\star\sta... | 4 | https://mathoverflow.net/users/11919 | 203746 | 98,125 |
https://mathoverflow.net/questions/203725 | 30 | I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem:
* Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into 1st and 2nd class citizens. Suppose that Hall's condition is satisfied for 1st class citizens. That is, for every set $... | https://mathoverflow.net/users/4354 | An unfair marriage lemma | In this answer I sketch an easy proof of your lemma and then give some references.
The easy proof uses Knaster's fixed point theorem:
THEOREM. Let $S$ be any set (finite or infinite) and let $\varphi:\mathcal P(S)\to\mathcal P(S)$ be an order-preserving map, i.e., $X\subseteq Y\implies\varphi(X)\subseteq\varphi(Y).... | 22 | https://mathoverflow.net/users/43266 | 203750 | 98,127 |
https://mathoverflow.net/questions/203751 | 6 | Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X\_t)\_{t\in T}$ defined by $X\_t = \langle G, t\rangle$, where $G$ is a standard Gaussian in $\mathbb{R}^n$ and $\langle\cdot, \cdot \rangle$ is the standard inner product. In other word, $\mathbb{E} X\_t... | https://mathoverflow.net/users/35733 | Does a Gaussian process shrink under a contraction map | It is true, and it follows from the following fact, sometimes referred to as Sudakov-Fernique inequality, sometimes as Slepian-Fernique lemma:
Assume that $(X\_t)$ and $(Y\_t)$ are two centered Gaussian processes. If $\|Y\_s - Y\_t\|\_2 \leqslant \|X\_s - X\_t\|\_2$ then $\mathbb{E} \sup Y\_t \leqslant \mathbb{E} \su... | 6 | https://mathoverflow.net/users/24953 | 203773 | 98,132 |
https://mathoverflow.net/questions/203732 | 19 | Let $K$ be a number field, and $\mathcal{O}\_K$ its ring of integers. Then there is an isomorphism
$$K\_0(\mathcal{O}\_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}\_K)$$
where $Pic(\mathcal{O}\_K)$ is the Picard group (or ideal class group) of $\mathcal{O}\_K$.
The cyclotomic trace defines a map
$$trc: K(\mathca... | https://mathoverflow.net/users/4649 | Can topological cyclic homology compute Picard groups? | Warning: I know nothing about this subject, but found the question interesting so decided to learn something about it. Approach the following with caution.
Consider the ring of integers $\mathbb{Z}[\tfrac{1+\sqrt{-15}}{2}] \subset \mathbb{Q}[\sqrt{-15}]$. This has class number 2, and the ideal
$$I:=(2, \tfrac{1+\sqrt... | 13 | https://mathoverflow.net/users/318 | 203789 | 98,135 |
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