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https://mathoverflow.net/questions/279037 | 2 | I want to ask a question that arises from reading [this paper](http://www.sciencedirect.com/science/article/pii/0022123676900173).
>
> Let $X$ be a locally compact space which is countable at infinity and let $\xi$ be a Radon measure on $X$. Suppose $V$ is a Hilbert space and there is a continuous and coercive bili... | https://mathoverflow.net/users/103376 | Two questions related to Dirichlet spaces and Sobolev spaces | For question 2, if you had $\Omega$ in place of $\bar{\Omega}$, the conditions you state would still be satisfied, but there are other interesting conditions that would not be. It would fail to be a *regular* Dirichlet space. In particular, you could not find a nice Markov process with state space $X = \Omega$ whose Di... | 2 | https://mathoverflow.net/users/4832 | 279041 | 123,677 |
https://mathoverflow.net/questions/279034 | 1 | Let $k$ be a field of characteristic zero and $X$ be a non-singular rationally connected variety over $k$. Let $L$ be a finite field extension of $k$. This induces a proper morphism $p:X\_L \to X\_k$. Is it true that for any torsion-free semi-stable sheaf $E$ on $X\_L$, the coherent sheaf $p\_\*E$ is semi-stable on $X\... | https://mathoverflow.net/users/43198 | Pushforward of semi-stable sheaves under finite field extension | Assuming what you mean is that $X$ is projective and $\mathcal E$ is Gieseker-semistable, the answer is yes.
>
>
> >
> > **Lemma.** Let $X$ be a projective $k$-scheme, and let $\mathcal E$ be a semistable sheaf on $X$. Then $\mathcal E\_{\bar k}$ is semistable on $X\_{\bar k}$.
> >
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> >
>
>
>
*Proof.* ... | 4 | https://mathoverflow.net/users/82179 | 279045 | 123,679 |
https://mathoverflow.net/questions/279054 | 0 | Let $G$ be a permutation group that acts on (say) $X=\{1,2,..,n\}$, and $H$ be a proper subgroup of $G$. Can one say anything precise about when the number of orbits of $H$ on $X$ will be equal to number of orbits of $G$ on $X$? In other words, is there a nice characterization of this situation, which perhaps gives som... | https://mathoverflow.net/users/102399 | number of orbits of a proper subgroup | I will write my actions on the left; for right actions, you would get $\mathrm{Stab}(x)H = G$ instead.
>
> Let $G$ be a group acting on a set $X$, let $x\in X$, and let $H$ be a subgroup of $G$. Then $Gx = Hx$ if and only if $H\mathrm{Stab}(x) = G$, where $\mathrm{Stab}(x) = \{g\in G\mid gx=x\}$ is the stabilizer o... | 3 | https://mathoverflow.net/users/3959 | 279061 | 123,683 |
https://mathoverflow.net/questions/278194 | 12 | This question is about Girard's system LU, presented in his paper [On the unity of logic](http://www.sciencedirect.com/science/article/pii/016800729390093S). Girard starts by giving a "modal" sequent calculus with two zones of both hypotheses and consequents, $\Gamma;\Gamma'\vdash \Delta';\Delta$, of which $\Gamma'$ an... | https://mathoverflow.net/users/49 | Is Girard's LU just an embedding of classical and intuitionistic logic into linear logic? | Sorry if my answer comes so late, maybe you already figured it out by yourself in the meantime, I hope this helps anyway.
I think the main misunderstanding is that the "non-chimeric" fragment of $\mathbf{LU}$ is *almost* a different presentation of linear logic, but not quite (of course it does not help that Girard c... | 7 | https://mathoverflow.net/users/45027 | 279073 | 123,689 |
https://mathoverflow.net/questions/279053 | 0 | Actually, in many works of probability theory/stochastic process, there is no explicit definition of randomness. Maybe because we think we can deduce the definition easily.
But in Kolmogorov complexity,randomness is defined strictly.
Now, my question is: could we have a formal definition of randomness in probabilit... | https://mathoverflow.net/users/14024 | Randomness defined in Kolmogorov complexity is identified with one in probability theory/stochastic process? | I would say the corresponding quantity for measuring randomness would be a quantity of information content, so typically Shannon entropy rate, i.e.,
$$
H(X)=\lim\_{n\rightarrow \infty}\frac{1}{n}H(X\_1,\ldots,X\_n)
$$
where the expectation for the entropy is taken over the joint pdf $\mathbb{P}(X\_1,\ldots,X\_n)$.
Se... | 2 | https://mathoverflow.net/users/17773 | 279079 | 123,691 |
https://mathoverflow.net/questions/278977 | 2 | The Kolmogorov Continuity theorem (see for example the [Wikipedia page](https://en.wikipedia.org/wiki/Kolmogorov_continuity_theorem)) lets us prove that a stochastic process $X\_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided we have a bound of the form
$$\mathbb E\left[d(X\_t,X\_... | https://mathoverflow.net/users/90954 | Kolmogorov continuity theorem and Holder norm | One can apply a deterministic result, called Garsia--Rodemich--Rumsey inequality, to estimate $\mathrm{E}[||X||^\alpha\_{\gamma;[0,T]}]$. Here is a particular form of this result, which is most convenient for us.
>
> For any $\alpha >1$, $\delta> 1/\alpha$, there is a constant $C(\delta,\alpha)$ such that for any $... | 9 | https://mathoverflow.net/users/8146 | 279085 | 123,693 |
https://mathoverflow.net/questions/279084 | 9 | Let $S$ be a Riemann surface with genus $g>0$. Let $M$ be the mapping class group of $S$. $Hom(\pi\_1(S),Gl(n, \mathbb{C}))$ is the representation space of fundamental group of $S$
**Question:** Is there an integer $n>1$, such that there exists an irreducible representation $\rho \in Hom(\pi\_1(S),Gl(n, \mathbb{C}))$... | https://mathoverflow.net/users/63996 | Mapping class group and representation of fundamental group of Riemann surfaces | There are counterexamples as soon as $g > 1$.
Let $n$ be the number of surjective homomorphisms $\pi\_1(S) \to A\_5$, up to $S\_5$-conjugacy. (We can see that $n \geq 1$ using the fact that $A\_5$ can be generated for two elements.)
Then there is a homomorphism $\pi\_1(S) \to A\_5^n$, where we take the product of ... | 10 | https://mathoverflow.net/users/18060 | 279094 | 123,696 |
https://mathoverflow.net/questions/279062 | 1 | I need to solve the following system of quasi-linear partial differential equations, where the unknowns $f(x,t), g(x,t)$ are smooth functions on $\mathbb R^2$,
$$ \dfrac{\partial}{\partial t}f=\dfrac{\partial}{\partial x}((f^2+g)f),$$
$$ \dfrac{\partial}{\partial t} g=\dfrac{\partial}{\partial x}((f^2+g)g),$$
I thi... | https://mathoverflow.net/users/20838 | Using method of characteristics to solve a system of first order quasilinear PDEs | The method of characteristics will tell you that, for any given solution $\bigl(f(x,t),g(x,t)\bigr)$, the characteristic curves are given by the foliations
$$
\mathrm{d}x + \bigl(3f(x,t)^2{+}2g(x,t)\bigr)\,\mathrm{d}t = 0
\quad\text{and}\quad
\mathrm{d}x + \bigl(f(x,t)^2{+}g(x,t)\bigr)\,\mathrm{d}t = 0.
$$
Thus, the ge... | 6 | https://mathoverflow.net/users/13972 | 279096 | 123,697 |
https://mathoverflow.net/questions/279099 | 4 | This question is motivated by recent works in quantum gravity, particularly in the analysis of the Sachdev-Ye-Kitaev (SYK) model. The SYK model is a one-dimensional quantum mechanical model which, in a certain limit, is dual to a two-dimensional black hole. (See [this paper](https://arxiv.org/pdf/1611.04650.pdf) for re... | https://mathoverflow.net/users/112368 | Ergodicity of the Form Factor in Random Matrix Theory | I don't think the late-time plateau in the spectral form factor is informative in the context of ergodicity of the ensemble of energy levels. The limit $K(t)\rightarrow 1$ for $t\rightarrow\infty$ and $N\rightarrow\infty$ seems a direct consequence of the central-limit theorem:
For $t>0$ the spectral form factor of a... | 5 | https://mathoverflow.net/users/11260 | 279107 | 123,699 |
https://mathoverflow.net/questions/278861 | 4 | Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold (which can depend on $d$ in some mild way, say logarithmically). How to find out whether the following true?
$\exists C,\epsilon>0$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:
$\left(... | https://mathoverflow.net/users/61472 | Implausible inequality | The "implausible" inequality is **true** for **any** $\epsilon$, for some $C$ and $K$, if $\space a>K\log(d)+K$.
Taking logarithms and then derivatives with respect to $t$ one can see that the unique maximum of
$L(t)\stackrel{\text{def}}{=}\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} t^{1+\e... | 1 | https://mathoverflow.net/users/2480 | 279128 | 123,705 |
https://mathoverflow.net/questions/279147 | 3 | Is there a connection on $\mathbb{R}^2 \setminus \{0\}$ for which all operators of parallel transports are in the form $$\begin{pmatrix}a&-b\\b&a \end{pmatrix}$$
but the parallel transport along circles with center at origin depends on the radius of the circle. That is two different circle have different parallel tra... | https://mathoverflow.net/users/36688 | Is there such a connection on the punctured plane? | Yes. Take the Levi-Civita connection of any conformal metric $g = e^{2u}(dx^2+dy^2)$ of positive curvature, say. Then, by (local) Gauss-Bonnet, the holonomy around any smooth closed loop $\gamma$ is of the above form with $a = \cos\theta(\gamma)$ and $b=\sin\theta(\gamma)$, where
$$
\theta(\gamma) = \int\_{\mathbb{R}^2... | 7 | https://mathoverflow.net/users/13972 | 279151 | 123,710 |
https://mathoverflow.net/questions/279153 | 1 | I am new to abelian varieties and became interested in the singular set of a subvariety in a complex torus.
The space of ample divisors (non necessarily smooth) in a complex torus $T^n$ seems quite large, as we can always move it, however, I don't have an example of a global hypersurface $M$ in $T^n$ with codim 1 si... | https://mathoverflow.net/users/12904 | On singular set of subvarieties in an abelian variety | Let $C$ be a smooth genus $2$ curve and let $J(C)$ be its Jacobian. Now, take a symmetric theta divisor $\Theta \subset J(C)$ and its translate $\Theta + x$, where $x$ is a $2$-torsion point. Finally, consider the isogeny of degree 2 $$f \colon J(C) \to A,$$
where $A$ is the quotient of $J(C)$ by the subgroup $\langle ... | 2 | https://mathoverflow.net/users/7460 | 279155 | 123,712 |
https://mathoverflow.net/questions/279154 | 2 | I'm working through Li's and Barlak's [Cartan Subalgebras and the UCT Problem](https://arxiv.org/pdf/1511.02697.pdf) but I'm stuck at one of the simpler proofs of the paper. On page 9 they deal with masas (maximal abelian subalgebras) of a C\*-algebra $A$ and look at the crossed products $A \rtimes\_\alpha \mathbb{Z}\_... | https://mathoverflow.net/users/64444 | Crossed products and unitaries implementing $\mathbb{Z}_n$-actions | The unitary exists by the definition of the crossed product. In the case of a $\mathbb{Z}/n\mathbb{Z}$-action you can think of an element of the crossed product as a linear combination
$$
\sum\_{n = 0}^{n-1} a\_n t^n
$$
with $a\_n \in A$ and where $t$ is a unitary (i.e. $t^\* = t^{-1}$), such that the following relatio... | 2 | https://mathoverflow.net/users/3995 | 279156 | 123,713 |
https://mathoverflow.net/questions/279150 | 18 | Apologies for the vagueness of question.
**Background**
[this thread](https://mathoverflow.net/a/45218/74739) has some nice examples of presheaves failing to be sheaves.
**Question**
Is there a generic way to measure "how badly" a presheaf fails at being a sheaf?
Something like an invariant that "counts", up ... | https://mathoverflow.net/users/74739 | Measuring a presheaf's failure to be a sheaf? | This answer is inspired by the Embedding Calculus (aka Manifold Calculus) of Weiss and Goodwillie. This is a framework for studying certain presheaves on manifolds. The idea is that sheafification of a presheaf is analogous to the linearization of a function. From this point of view, sheafification is just the first in... | 24 | https://mathoverflow.net/users/6668 | 279157 | 123,714 |
https://mathoverflow.net/questions/278212 | -1 | Let V be a two dimensional vector space over k. Consider the set of isomorphism classes of all rank 1 $k[x]/x^2$-submodules of $V\otimes k[x]/x^2$. Does this set has a variety structure? Is it tangent bundle of $\mathbb{P}^1$?
| https://mathoverflow.net/users/nan | tangent bundle of $\mathbb{P}^1$ | It may be classical to consider varieties as sets, but I think it is having counterproductive effects on communication in this case. The set you are considering is now known as the set of $k$-points of the variety.
For any commutative ring $k$, the tangent bundle of $\mathbb{P}^1\_k$ can be seen as a functor that sen... | 1 | https://mathoverflow.net/users/121 | 279165 | 123,718 |
https://mathoverflow.net/questions/279140 | 3 | Is there anything known about the maximum number of simple-polygonal Hamilton cycles that a straight-line drawing of a Hamiltonian graph can have?
Put differently, if the vertices of a Hamilton graph are mapped to points of the euclidean plane and the edges to straight-line segments connecting the images of their ad... | https://mathoverflow.net/users/31310 | Number Associated with Straight-line Drawings of Hamiltonian Graphs | It is known that the number of non-crossing spanning cycles (called "simple polygonalizations") of $n$ points in the plane can be as low as $1$ (for points in convex position) and as high as $4.64^n$, and that it is never higher than $94^n$. See:
[On the Number of Crossing‐Free Matchings, Cycles, and Partitions](http... | 5 | https://mathoverflow.net/users/440 | 279174 | 123,720 |
https://mathoverflow.net/questions/279160 | 3 | For the definition of a semicomputable real, see *An Introduction to Kolmogorov Complexity and its Applications* by Li and Vitanyi (1997). In fact, it is not true that every Cauchy sequence for completion of rationals is semicomputable, we can not complete rationals by semicomputable Cauchy sequences.
My question:
... | https://mathoverflow.net/users/14024 | Is any Cauchy sequence for completion of rational semicomputable? | Constructions of real numbers broadly fall into two classes: Dedekind-style completions by cuts ("Dedekind reals"), and Cauchy-style completions by Cauchy sequences ("Cauchy reals"). There is a well understood theory of these in constructive mathematics, and the connections between the different constructions are well ... | 7 | https://mathoverflow.net/users/1176 | 279178 | 123,721 |
https://mathoverflow.net/questions/279173 | 12 | Excuse me for the concern, but I want to ask you a question.
In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science of physics (see [1]). The scripts were interesting but so far I could not find any article where $q$-mathematics was ap... | https://mathoverflow.net/users/110748 | Is there any published physics article where $q$-mathematics is applied? | There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-deformations of simple harmonic oscillator(s) and angular momentum algebras to the development of quantum groups and their applica... | 23 | https://mathoverflow.net/users/85967 | 279184 | 123,723 |
https://mathoverflow.net/questions/279149 | 5 | Let $R$ be a ring (not commutative in general) with identity and let
$\Psi(x) = x^m-\sum\_{j=0}^{m-1}\psi\_jx^j$ be a monic polynomial over $R$. I want to construct a ring extension $K$ of $R$, which contains a root of $\Psi(x)$.
One construction follows from [non-commutative Hamilton-Caley Theorem](https://mathoverfl... | https://mathoverflow.net/users/85489 | Where can we find polynomial's root? | No, $R[x]/A$ does not always have a root.
Let $R=\mathbb{Z}\langle a,b\rangle$ be the free ring on two noncommuting elements $a$ and $b$. Let $\Psi=x^2+a\in R[x]$, and let $M$ and $A$ be the objects you defined.
In this case, $A=0$. Suppose $f=r\_nx^n+\cdots+r\_0\in A$. In particular, there is some $g=s\_n x^n+\cdo... | 3 | https://mathoverflow.net/users/112641 | 279189 | 123,726 |
https://mathoverflow.net/questions/279193 | 4 | I am reading the survey of the relationships between metrics of distributions (see <https://arxiv.org/pdf/math/0209021.pdf> for the paper).
The general results show that for general distributions, we cannot upper bound the total variation by Wasserstein distance. Many answers in MO give the same intuitive counterexampl... | https://mathoverflow.net/users/113059 | Upper bound total variation by Wasserstein distance for continuous distance | No. One should realize that the transportation and the total variation distances metrize two quite different topologies. Even if the measures are equivalent (i.e., absolutely continuous with respect to each other), one can still easily have examples when the transportation distance is arbitrarily close to 0, whereas th... | 9 | https://mathoverflow.net/users/8588 | 279196 | 123,727 |
https://mathoverflow.net/questions/279198 | 0 | Let $F$ be a metrizable locally convex space (you may assume it is a Banach space), and let $E$ be a complete locally convex space (you may assume it is a Frechet space). Let $T$ be a continuous linear map from $F$ into $E$ and let $H\subset E^{\*}$ be such a subspace that $\overline{T^\*H}^{F^\*}=\overline{T^\*E^\*}^{... | https://mathoverflow.net/users/53155 | Criterion for weak compactness | If the condition $\overline{T^\*H}^{F^\*}=\overline{T^\*E^\*}^{F^\*}$ refers to the weak$^\*$ topology on $F^\*$ a counterexample is provided by any non-reflexive Banach space $X$ with $E=F=X^\*$ (with the dual Banach space norm), $T=$ id, $H=X\subseteq X^{\ast\ast}$ and $B$ the unit ball of $X^\*$:
Since $X$ is $\sigm... | 1 | https://mathoverflow.net/users/21051 | 279202 | 123,729 |
https://mathoverflow.net/questions/261895 | 4 | Let $\mathfrak g$ be a complex simple Lie algebra, $l$ be a natural number, and $V=V^l(\hat g)$ be the vertex operator algebra of the affine Lie algebra $\hat{\mathfrak g}$ at level $l$. We know that $V$ can always give rise to a conformal net $\mathcal A\_V$ (constructed say by integrating the loop algebra $L\_I\mathf... | https://mathoverflow.net/users/86652 | Strong additivity of the conformal net of an affine simple Lie algebra | In Section 4.C of my paper <https://arxiv.org/pdf/1302.2604.pdf>, I discuss various aspects of the loop group conformal nets, with a couple of pointers to the literature.
My understanding is that section IV.1 of Toledano-Laredo's PhD works equally well for all compact Lie groups $G$.
Actually, if you look at the wo... | 2 | https://mathoverflow.net/users/5690 | 279221 | 123,734 |
https://mathoverflow.net/questions/279222 | 3 | Since the natural logarithm, i.e. with base $e$, is very commonly used in research papers and that both $\ln(x)$ and $\log(x)$ are used to denote it, it is natural\* to ask which of these notations to use when preparing a paper. The fact that both are used in literature concerning the same topics gives rise to unnecess... | https://mathoverflow.net/users/103722 | $\log(x)$ or $\ln(x)$ to denote the natural logarithm in research papers? | In number theory, the notation $ \log $ is commonly used, especially when asymptotics are considered. One also frequently uses the notation $ \log\_{k} $ for the $ k $ -th iterate of this function. Indeed the natural logarithm is essentially the only one that matters. This may not be true for other subfields of mathema... | 4 | https://mathoverflow.net/users/13625 | 279226 | 123,737 |
https://mathoverflow.net/questions/279064 | 11 | I find this question interesting, but need to get it out of my system: is the space of connections (modulo gauge) on a compact four-manifold paracompact, in the Sobolev topology?
If so, I believe it would admit partitions of unity, which would surely make life easier in gauge theory. But I haven't seen the experts ma... | https://mathoverflow.net/users/113568 | Is the space of connections modulo gauge equivalence paracompact? | Yes, the space of gauge orbits of connections is paracompact (even when you the use Fréchet topology).
First, the space of all connections is paracompact since it is an affine space modelled on a nuclear Fréchet space (and/or it is metrisable). Narasimhan & Ramadas (Geometry of SU(2) Gauge Fields) showed that the act... | 8 | https://mathoverflow.net/users/17047 | 279228 | 123,738 |
https://mathoverflow.net/questions/279230 | 5 | Let $\{K\_i\}$ be a sequence of convex compact $n$-dimensional subsets in a Euclidean space $\mathbb{R}^n$. Assume $\{K\_i\}$ converges in the Hausdorff metric to a convex compact set $K$ which is also $n$-dimensional.
Consider the sequence of boundaries $\{\partial K\_i\}$ equipped with the induced intrinsic (!) me... | https://mathoverflow.net/users/16183 | Hausdorff vs Gromov-Hausdorff convergence of convex hypersurfaces | The reference is Lemma 10.2.7 in [A course of metric geometry](http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf) by Burago-Burago-Ivanov. They do it in 3d but it does not matter. The main point is that if two convex bodies are Hausdorff close, then one can blow up one of them by slight dilation to contain th... | 10 | https://mathoverflow.net/users/1573 | 279234 | 123,739 |
https://mathoverflow.net/questions/279240 | 2 | Let $V$ be a set and let $V^V$ denote the set of all functions $f:V\to V$. Suppose that $F\subseteq V^V$. Let $[V]^2 = \big\{\{x,y\}: x, y\in V \land x\neq y\big\}$. We say $E\subseteq [V]^2$ is *$F$-compatible* if all members of $F$ are [graph homomorphisms](https://en.wikipedia.org/wiki/Graph_homomorphism) from $(V,E... | https://mathoverflow.net/users/8628 | Graph structures compatible with a collection of functions | Yes.
Each $f:V\to V$ induces a map $\bar f:V^2\to V^2$ in the natural way.
Suppose an edge $e\in [V]^2$ appears in an $F$-compatible graph $(V,E)$. Then for all $f\in F$, $\bar f(e)$ must be in $E$ as well, as must $(\bar f\circ\bar f)(e)$ and so on. The sequence $(\bar f^{\circ n}(e))\_n$ either eventually squashe... | 2 | https://mathoverflow.net/users/112641 | 279252 | 123,747 |
https://mathoverflow.net/questions/279241 | 4 | For normality, see <https://en.wikipedia.org/wiki/Normal_number>. For random number/sequence, see <https://en.wikipedia.org/wiki/Algorithmically_random_sequence>.
Now, is there any number that is normal in every bases $b > 1$ except random numbers (or numbers expansion of which is algorithmically random sequence)?
... | https://mathoverflow.net/users/14024 | Existence of normal number except random numbers | Computable, absolutely normal numbers do actually exist. See
V. Becher, S. Figueira: *[An example of a computable absolutely normal number](https://doi.org/10.1016/S0304-3975(01)00170-0)*, Theoretical Computer Science **270** (2002), 947-958.
| 16 | https://mathoverflow.net/users/7460 | 279256 | 123,749 |
https://mathoverflow.net/questions/279239 | 3 | An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^\*M)\otimes TM$. One would think that this affine bundle is an associated bundle to the second order (co-)frame bundle $F^2(M) \to M$. Hence there should be a natural af... | https://mathoverflow.net/users/nan | Affine connections as equivariant maps | There is a construction that works for connections on arbitrary principal bundles (and not just for the frame bundle):
Let $P$ be a principal $G$-bundle over a $n$-dimensional manifold $M$. The space of principal connections on $P$ can be identified with sections of an affine bundle $QP \to M$. In order to realize $QP$... | 1 | https://mathoverflow.net/users/17047 | 279268 | 123,753 |
https://mathoverflow.net/questions/279258 | 2 | Consider the differential equation
$$ m \ddot{x} + k \dot{x} = - W\_t x $$
where
* $m$ and $k$ are nonnegative.
* $x\_t \in \mathbb{R}^n$
* $W\_t$ is a matrix that satisfies $$ \alpha I \succeq W\_t \succeq \beta I > 0 ~~\mbox{ for all } t$$
My question is: can we conclude that $x\_t$ bounded?
If $W\_t= W$... | https://mathoverflow.net/users/113661 | Boundedness of particle motion with time-varying force | This does not follow. We can find a counterexample in dimension $n=1$ (and let's also set $m=1$). If we write $x=ye^{-kt/2}$, then $y$ solves
$$
-y'' - W(t) y = -\frac{k^2}{4} y ,
$$
and I want to interpret this as a 1D Schrödinger equation at energy $E=-k^2/4$. For a periodic potential, the spectrum has band structure... | 1 | https://mathoverflow.net/users/48839 | 279271 | 123,754 |
https://mathoverflow.net/questions/279231 | 12 | In his book, "The Strange Logic of Random Graphs", Joel Spencer describes the "Dance Marathon" problem:
>
> Imagine $n$ couples at a Dance Marathon. Each dance each couple remains standing with independent probability one half. A couple that does not remain standing is removed from the competition. A couple wins th... | https://mathoverflow.net/users/8938 | The dance marathon problem | I'll lay out the starting steps; I hope that after that it won't be much work for you to fill in on your own.
To be clear, the process is that there are a succession of dances. At the start, $n$ couples are dancing. In each round, each couple is (independently) eliminated with probability $1/2$. So the probability of... | 8 | https://mathoverflow.net/users/297 | 279273 | 123,755 |
https://mathoverflow.net/questions/279204 | 1 | Let $X$ be a projective, noetherian $k$-scheme for an algebraically closed field $k$ of characteristic zero. Let $Y\_1,...,Y\_r$ be locally closed subschemes (open subschemes of closed subschemes) of $X$. Does the scheme structure on $X$ necessarily induce a scheme structure on $Y\_1 \cup ... \cup Y\_r$?
| https://mathoverflow.net/users/43198 | Is finite union of locally closed subscheme, a scheme | You just discovered *constructible sets*!
It is really easy to give counter-examples to your suggestion (as *Ja ok* already has), but here is a general idea:
Take your favorite locally closed but neither open nor closed subscheme of your favorite irreducible scheme. Then prove (as a homework) that its complement is ... | 5 | https://mathoverflow.net/users/10076 | 279274 | 123,756 |
https://mathoverflow.net/questions/279277 | 2 | Let G be a finite group and we know its group table is a Latin square of order |G|. Now let H be any subgroup of G of index n. Then we can form G/H which is a collection of left cosets. My question is, is there any natural construction to get a Latin square of order n from G/H? If H is normal in G it is clear since G/H... | https://mathoverflow.net/users/110997 | How to get Latin squares from a finite group and a subgroup | I don't know of any general construction. Perhaps the most 'natural' examples are loop transversals: Given a left transversal $X$ to $H$ in $G$ with $1 \in X$, define a binary operation on $X$ by setting $x\*y$ to be the unique element of $X \cap xyH$. Then $\*$ is left-cancellative; if $X$ is in fact a left transversa... | 3 | https://mathoverflow.net/users/4053 | 279287 | 123,760 |
https://mathoverflow.net/questions/279020 | 4 | Before asking my question, consider elliptic curves over complex numbers.
Let $E\_{\tau}=\mathbb{C}/{\Lambda\_{\tau}}$ be an elliptic curve over $\mathbb{C}$, where $\Lambda\_\tau=\mathbb{Z} \oplus \tau \mathbb{Z}$ with $\text{Im} (\tau) > 0$.
Let $N>3$ be a prime number. Then,
the number of cyclic subgroups of o... | https://mathoverflow.net/users/46108 | Atkin-Lehner involutions on modular curves over finite fields | Even for $N=2$ it is not true that $E/C\_i$ is necessarily isomorphic to $E$. In fact, the isogeny graph consisting of supersingular elliptic curves in char $p$ as vertices and $2$-isogenies as edges is connected, so as soon as there are more than one supersingular elliptic curve in char $p$, there will be an $i$ with ... | 2 | https://mathoverflow.net/users/2290 | 279291 | 123,762 |
https://mathoverflow.net/questions/279284 | 2 | It is well known that the [von Neumann universes](https://en.wikipedia.org/wiki/Von_Neumann_universe) $V\_{\alpha}$ is a model of ZF(C) when $\alpha$ is an inaccessible cardinal. In the following let $V$ be such a model of ZF(C). It is also well known (see corollary 5.3 of Set Theory, The Third Edition, by Thomas Jech)... | https://mathoverflow.net/users/113675 | Existence of regular cardinals larger than an arbitrary cardinal in von Neumann universes without axiom of choice | Actually, if you assume that every set is inside a universe, then you can get something slightly weaker, but you do get a class of inaccessible cardinals which are regular. So this is already something.
Patterns of singular cardinals can be difficult to obtain sometimes, but we have no reason to believe that they ar... | 3 | https://mathoverflow.net/users/7206 | 279292 | 123,763 |
https://mathoverflow.net/questions/279293 | 9 | Consider the following 2-category:
• It objects are concrete categories, i.e., categories equipped with a faithful functor to $Set$.
• A 1-morphism between $(C\_1,U\_1)$ and $(C\_2,U\_2)$ consist of a functor $F:C\_1\to C\_2$ and a natural transformation $z:U\_1\Rightarrow U\_2\circ F$.
• Its 2-morphisms are the... | https://mathoverflow.net/users/5690 | Category of concrete categories | **Concrete functor** is established in the literature for the related notion where the natural transformation is an isomorhpism (see e.g. Porst 1996 [*Concrete Categories Are Concretely Equivalent if…*](https://link.springer.com/content/pdf/10.1007/BF00124121.pdf)) — i.e. the sub-2-category of the slice 2-category of *... | 10 | https://mathoverflow.net/users/2273 | 279298 | 123,765 |
https://mathoverflow.net/questions/278828 | 7 |
>
> Question 1: What is a complete classification of all positive integers $m,n$ with the following property:
>
>
> **There is a continuous map $f:S^n \to \mathbb{C}P^m$ such that $f$ maps antipodal points to orthogonal lines. Namely for every $x\in S^n$ we have $\;f(x) \perp f(-x)$. Here the later perpendicularity... | https://mathoverflow.net/users/36688 | Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $ | Such a map $S^n\to \mathbb CP^m$ exists if and only if either $n<2m$ or $n=2m=2$.
To see this, first note that such a map is the same as a $\mathbb Z/2$-equivariant map from $S^n$ to a certain subspace of $\mathbb CP^m\times \mathbb CP^m$, namely the space of pairs $(L,M)$ such that $L\perp M$. Now note that the latt... | 11 | https://mathoverflow.net/users/6666 | 279302 | 123,766 |
https://mathoverflow.net/questions/279306 | 9 | Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}\_Y)$ depend on $Y$?
I'm sure that the answer is 'no' if $X$ is normal, but I can't find a reference. Is it true in g... | https://mathoverflow.net/users/81928 | Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution? | **Edit. This follows from the Elkik-Fujita Vanishing Theorem.** There is a more general vanishing theorem due to Elkik and Fujita. One version of this theorem (where I read the theorem) is Theorem 1.3.1 of the following article.
MR0946243 (89e:14015)
Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji
Introdu... | 8 | https://mathoverflow.net/users/13265 | 279310 | 123,768 |
https://mathoverflow.net/questions/279317 | 8 | To celebrate my birthday, I like to find interesting number theoretic
properties of my new age. My upcoming 61st birthday was challenging, but then
I noticed that $61 = 5^2 + 6^2 = 5^3 - 4^3$, the sum of two consecutive squares
and the difference of two consecutive cubes. I wondered what other numbers
had this property... | https://mathoverflow.net/users/22344 | Sum of two consecutive squares equals difference of two consecutive cubes | There is the following sequence of positive solutions (according to Mathematica): $a\_n=\frac{1}{8} \left(\left(\sqrt{6}-2\right) \left(2 \sqrt{6}+5\right)^n-\left(\sqrt{6}+2\right) \left(5-2 \sqrt{6}\right)^n-4\right),$ and
$b\_n=\frac{1}{12} \left(-\left(\sqrt{6}-3\right) \left(2 \sqrt{6}+5\right)^n+\left(\sqrt{6}+3... | 9 | https://mathoverflow.net/users/41145 | 279320 | 123,770 |
https://mathoverflow.net/questions/279329 | 2 | Let $G=Sl\_2(\mathbb{F}\_p)$ and $M= \mathbb{F}\_p[x\_1,x\_2]$, where $p$ is a prime.
$M$ is a $G$-module with $(A\cdot x\_1, A\cdot x\_2)=(x\_1,x\_2)\cdot A, (\forall) A \in Sl\_2(\mathbb{F}\_p)$.
I have to show that
$$M^G = \mathbb{F}\_p[x\_1\cdot v, x\_1^{p\cdot(p-1)}+ v^{p-1}],$$
where
$$v=x\_2\cdot(x\... | https://mathoverflow.net/users/113626 | How to prove that $M^G=\mathbb{F}_p[x_1\cdot v, x_1^{p\cdot(p-1)}+ v^{p-1}]$? | Let $R$ be the ring that you think is equal to $M^G$. You need to check that $R$ is fixed by $G$ and that $M$ is a free module of rank equal to $|G|$ over $R$. Then Galois theory tells you that the field of fractions $QR$ is $(QM)^G$. Moreover, as $R$ is a polynomial ring it is a unique factorisation domain and so is i... | 2 | https://mathoverflow.net/users/10366 | 279332 | 123,775 |
https://mathoverflow.net/questions/279328 | 13 | The symmetric group $S\_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.
So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).
I'd like to find the decomposition into irreps of it. As I'm a physicist, I'm first interested in the simple cases $... | https://mathoverflow.net/users/113692 | Tensor power of the natural representation of Sn | This question was recently completely solved in [this paper](https://arxiv.org/abs/1605.06543).
As explained on page 15 of the paper, letting $v\_1,...,v\_n$ be a basis for $V$, the standard basis vectors $v\_{i\_1} \otimes \cdots \otimes v\_{i\_p}$ for $V^{\otimes p}$ naturally correspond to partitions of the set $\... | 16 | https://mathoverflow.net/users/33089 | 279333 | 123,776 |
https://mathoverflow.net/questions/279323 | 1 | I have been reading about moment problem and I have been curious about the following question.
What is the motivation for studying the Hamburger moment problem(one dimensional moment problem?
I have been asking myself out of curiosity, what is gained by people who considered this case? or let me say how can I exp... | https://mathoverflow.net/users/104899 | Why study the moment problem in one dimensional case( Hamburger moment problem) | It has many applications, especially in probability: e.g. you want to know if there is a probability distribution satisfying some condition on its moments.
Or you know the moments of a random variable match those of a certain distribution, and you want to know if that implies this is the distribution of the random vari... | 3 | https://mathoverflow.net/users/13650 | 279341 | 123,780 |
https://mathoverflow.net/questions/279347 | 1 | Let $R$ be a commutative ring with unity and let $S$ be a multiplicatively closed subset of $R$ such that $S$ contains no zero divisor . So the canonical map $f : R \to S^{-1}R$ is invective , hence w.l.o.g. , let us assume $R$ is a subring of $S^{-1}R$ . Now if for every $f :R \to R , \exists \hat f (x) \in (S^{-1}R)[... | https://mathoverflow.net/users/nan | Functions on rings and polynomials with coefficients in a certain kind of localisation | Yes (assuming $R$ is nonzero).
Suppose $a,b\in R\setminus\{0\}$ satisfy $ab=0$. Suppose further that $f=\frac{r\_n}{s\_n}x^n+\cdots+\frac{r\_1}{s\_1}x+\frac{r\_0}{s\_0}\in (S^{-1}R)[x]$ satisfies $f(0)=1$ and $f(a)=f(b)=0$. In particular $r\_0/s\_0=1$. Then $0=f(a)f(b)=f(a)+f(b)-1=-1$, because all terms in the produc... | 3 | https://mathoverflow.net/users/112641 | 279350 | 123,783 |
https://mathoverflow.net/questions/279335 | 0 | Let $G$ be a undirected graph with $n$ many vertices and $m$ many edges.
Let us define $q = \Theta \big(\frac{n} {\log n}\big)$, now let us call a vertex $v$ **big** if degree$(v\_i) \ge \frac{m}{q}$.
**Question :** How many big vertices will be there in a graph ?
I know the loose upper bound is $q$, Is it tight ?... | https://mathoverflow.net/users/nan | Tight upper bound on the number of high degree vertices | Suppose there are $b$ big vertices. Then there are at least $\frac{mb}{2q}$ edges incident to these vertices. Hence,
$$\frac{mb}{2q} \leq m$$
implying that $b\leq 2q$.
To get an example with $2q$ big vertices, let them form an $\frac{m}{q}$-regular graph and the other $n-2q$ vertices be isolated.
---
**UPDATE**... | 3 | https://mathoverflow.net/users/7076 | 279355 | 123,785 |
https://mathoverflow.net/questions/278619 | 14 | Why are so many algebraists nowadays interested in cluster algebras?
(This is a rewording of one half of the closed question [Cluster algebras and teichmuller theory](https://mathoverflow.net/questions/278574/cluster-algebras-and-teichmuller-theory).)
| https://mathoverflow.net/users/39082 | Applications of cluster algebras | One reason is that cluster algebras have motivated many recent developments in the representation theory of associative algebras. There is a lot one can say about this, so I will try to just give an overview of some of the key ideas, and suggest further reading. I recommend Keller's survey article (<https://arxiv.org/a... | 10 | https://mathoverflow.net/users/21483 | 279380 | 123,793 |
https://mathoverflow.net/questions/279374 | 4 | Suppose $A,A\_1,\ldots,A\_{n-2}$ (resp. $B$) are (resp. is) real positive-definite (resp. arbitrary) symmetric $n\times n$ matrices and denote by $D(\cdot,\ldots,\cdot)$ the mixed discriminant. We have the following well-known Aleksandrov-Fenchel inequality
\begin{equation}\label{e}
D(A,B,A\_1,\ldots,A\_{n-2})^2\geq... | https://mathoverflow.net/users/36974 | The Aleksandrov-Fenchel inequality of mixed discriminants for Hermitian matrices | [The Van der Waerden Conjecture for Mixed Discriminants](https://arxiv.org/abs/math/0406420), by Leonid Gurvits (2004), proves a generalized Alexandrov-Fenchel inequality for semidefinite $n\times n$ Hermitian matrices:
Theorem 5.2, with equation (21), that for the special case $\alpha=(1,1,\ldots,1)$, $\alpha^{(1)}... | 1 | https://mathoverflow.net/users/11260 | 279383 | 123,794 |
https://mathoverflow.net/questions/279299 | 7 | A minimally strongly connected digraph (MSC) is [strongly connected](https://en.wikipedia.org/wiki/Strongly_connected_component) (SC), while removal of any arc destroys this. That is, between any two vertices a, b there exists a directed path from a to b, while removal of any arc (a,b) renders b unreachable from a.
W... | https://mathoverflow.net/users/113448 | Graph isomorphism problem for minimally strongly connected digraphs | Isomorphism of MSC digraphs is *isomorphism-complete*. Consider two connected undirected graphs $G,H$ with no vertices of degree 1. It is routine to see that connectivity and minimum degree at least 2 won't help you to determine whether $G$ and $H$ are isomorphic.
Now convert $G,H$ to digraphs $G',H'$ by replacing ea... | 3 | https://mathoverflow.net/users/9025 | 279386 | 123,795 |
https://mathoverflow.net/questions/279057 | 11 | I apologize if this is too obvious, but I figure it must have a quick answer.
Are there open subgroups $\Gamma\le SL\_2(\widehat{\mathbb{Z}})$ which are conjugate in $GL\_2(\widehat{\mathbb{Z}})$, but not conjugate in $SL\_2(\widehat{\mathbb{Z}})$?
| https://mathoverflow.net/users/88840 | Are there open subgroups of $SL_2(\widehat{\mathbb{Z}})$ which are $GL_2(\widehat{\mathbb{Z}})$-conjugate, but not $SL_2$-conjugate? | Yes, there are indeed such subgroups. Since the open subgroups are exactly the congruence subgroups, it suffices to find two subgroups of $\mathrm{SL}\_2(\mathbf{Z}/N\mathbf{Z})$ which are conjugate in $\mathrm{GL}\_2(\mathbf{Z}/N\mathbf{Z})$ but not in $\mathrm{SL}\_2(\mathbf{Z}/N\mathbf{Z})$. The following Magma code... | 11 | https://mathoverflow.net/users/6506 | 279398 | 123,800 |
https://mathoverflow.net/questions/269354 | 18 | I would like to know which motivic cohomology groups of complex numbers are non-zero and ("better") non-torsion, i.e., for which $(i,j)$ the $i$th cohomology of the complex ${\mathbb{Q}}(j)$ over $\mathbb{C}$ is not zero. I would like to know both which of these groups are known to be non-zero (and also known to be "la... | https://mathoverflow.net/users/2191 | Which motivic cohomology groups of complex numbers are non-torsion? | This is going to be a slightly extended explanation, I apologize. The short version is basically that little is actually known (and even that is hard to prove), but conjecturally everything permitted by Beilinson-Soulé vanishing should be infinite-dimensional.
**What is known unconditionally** (I bet everybody knows... | 13 | https://mathoverflow.net/users/50846 | 279403 | 123,801 |
https://mathoverflow.net/questions/279407 | 3 | Consider the interval $I=(0,1)$ and let $f,g$ be two linearly independent continuous functions on $[0,1]$.
I am asking if there is a continuous function $h$ such that
$$\int\_0^1 h(s) f(s) ds=0$$
$$\int\_0^1 h(s) g(s) ds \neq 0$$
and $h(0)=h(1)=0$.
If it were only the first two conditions, then Gram-Schmidt o... | https://mathoverflow.net/users/112877 | Prove existence of continuous function on $(0,1)$ with special properties | I guess you can just work with the measure $d\mu(x) = x(1-x)\, dx$. By Gram-Schmidt, there is a function $\tilde{h}$ such that
$$
\int\_0^1 \tilde{h}(x) \, f(x) \, x(1-x) \, dx =0,
$$
$$
\int\_0^1 \tilde{h}(x) \, g(x) \, x(1-x) \, dx \neq 0.
$$
The function $\tilde{h}$ is explicit, using the scalar product given by $\m... | 2 | https://mathoverflow.net/users/6129 | 279409 | 123,803 |
https://mathoverflow.net/questions/279219 | 3 | Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
Is it true that any such hyperbolic 3-manifold of finite volume can be obtained as $S^{3}-K$ where $K$ is some knot?
... | https://mathoverflow.net/users/103287 | hyperbolic 3-manifold of finite volume | One can regard the [geometrization theorem](https://en.wikipedia.org/wiki/Geometrization_conjecture) as a classification of hyperbolic 3-manifolds of finite volume. These are the interiors of ($\mathbb{P}^2$-)irreducible compact 3-manifolds $M$ with incompressible boundary, Euler characteristic $=0$, and atoroidal (eve... | 8 | https://mathoverflow.net/users/1345 | 279417 | 123,808 |
https://mathoverflow.net/questions/279426 | 13 | Let $S\_6$ be the symmetric group on 6 letters and let $\alpha \colon S\_6 \to S\_6$ be an outer automorphism (note that $S\_6$ is the only permutation group that has an outer automorphism and that $\mathrm{Out}(S\_6) \cong \mathbb{Z}/2\mathbb{Z}$). For any irreducible representation $\rho \colon S\_6 \to \mathrm{GL}(V... | https://mathoverflow.net/users/4428 | Outer automorphism action on representations of $S_6$ | First, notice that this operation preserves the dimension of the representation, this already considerably restricts things: the dimensions of the irreps. are 1,1,5,5,5,5,9,9,10,10,16. The trivial rep. is fixed, thus the sign rep. must also be. The 16-dimensional rep. $V\_{(3,2,1)}$ is also fixed.
Next, since $\alpha... | 4 | https://mathoverflow.net/users/33089 | 279433 | 123,814 |
https://mathoverflow.net/questions/279401 | 2 | The simple continued fraction is in the form
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x\_i$can be computed by computable function $x\_i=f(i), i\in \mathbb{N},$ and, $i$ is the $i$th coefficients.
Is there any function by which to compute coefficients of co... | https://mathoverflow.net/users/14024 | Reference request for function by which to compute coefficients of continued fraction of algebaic number | This paper may be worth a look:
E. Bombieri and A. J. van der Poorten. Continued fractions of algebraic numbers. Computational algebra and number theory (Sydney, 1992), pp. 137–152. Kluwer Acad. Publ., 1995.
Also, R.P. Brent, Alfred J. van der Poorten, Herman J.J. te Riele: A comparative study of algorithms for co... | 5 | https://mathoverflow.net/users/3684 | 279434 | 123,815 |
https://mathoverflow.net/questions/279448 | 2 | Let $E$ be the elliptic curve over $\mathbb{Q}$ defined by
$$
y^2=x^3-1.
$$
Let $p$ be an odd prime congruent $-1$ modulo $3$. Then,
this curve becomes supersingular after ''reduction'' modulo $p$, also denoted by $E$. Note that the group of automorphisms of $E$, $\text{Aut} (E)$, is cyclic of order $6$. Let $\rho$ b... | https://mathoverflow.net/users/46108 | Supersingular elliptic curves with automorphisms of order 6 | No. Frobenius switches those two subgroups.
To prove this, consider the eigenvalue of $\rho$ on each of those subgroups, which is one of the two sixth roots of unity in $\mathbb F\_q$. Because $\operatorname{Frob}\_p(\zeta)=\zeta^{-1}$ for $\zeta$ a third root of unity, we have $\operatorname{Frob}\_p (\rho)=\rho^{-... | 6 | https://mathoverflow.net/users/18060 | 279456 | 123,822 |
https://mathoverflow.net/questions/279419 | 2 | I read the article "Defomrations of algebras in noncommutative geometry" by Schedler.
In Definition 3.7.9. he gives the definition of Calabi-Yau algebra of dimensi on d as algebras that are homological smooth with $HH^{\*}(A,A \otimes\_k A)=A[-d]$ as graded algebras.
My question: For finite dimensional algebras, is t... | https://mathoverflow.net/users/61949 | Calabi-Yau algebra for finite dimensional algebras | It is not particularly productive to think about finite-dimensional Calabi–Yau algebras, at least if you are grading everything in degree $0$ (and possibly even if you have a more interesting grading—I have never seriously thought about this case), since they are all semi-simple.
Let $A$ be a non-zero finite-dimensio... | 1 | https://mathoverflow.net/users/21483 | 279457 | 123,823 |
https://mathoverflow.net/questions/279462 | 7 | I think the following problem is difficult, any ideas for solution are welcome.
Find all integer solutions to $y^2=x^5+4.$
Is it true that the only solutions are $(x, y) = \{( 2,-6), ( 2,6),(0,2), (0,-2) \}.$?
| https://mathoverflow.net/users/110748 | Find all integer solutions to $y^2=x^5+4.$ Is it true that the only solutions are $(x, y) = \{( 2,-6), ( 2,6),(0,2), (0,-2) \}.$ | Yes, these are the only examples.
Either $x$ is zero (your $(0, \pm 2)$ example) or not; assume the latter. Rewrite your equation as $(y - 2)(y + 2) = x^5$, and appeal to unique factorisation. Then $y \pm 2$ are either both fifth powers, or one is of the form $4a^5$ and the other is of the form $8b^5$. The former cas... | 15 | https://mathoverflow.net/users/39521 | 279470 | 123,826 |
https://mathoverflow.net/questions/279471 | 3 | Is it true that any finite dimensional division algebra over a pseudo-algebraically closed field is trivial? We know that this is true for algebraically closed field.
| https://mathoverflow.net/users/43198 | Finite dimensional division algebra over pseudo-algebraic closed field | The fact that every finite dimensional division algebra is trivial is not only true over an algebraically closed field but it is in fact equivalent to the field being algebraically closed. Remember that (extension-)fields are just a special case of division algebras. Thus, if it would be true over pseudo-algebraically ... | 6 | https://mathoverflow.net/users/109932 | 279472 | 123,827 |
https://mathoverflow.net/questions/279461 | 7 | I would like to know about the literature concerning the group of outer automorphisms of the Lie algebra $\mathfrak{sl}(2,R)$. This question is addressed in different places in a contradictory way. In certain works, e.g.
M.A. Farinati and A.P Jancsa,
Three dimensional real Lie bialgebras,
Revista de la union matemat... | https://mathoverflow.net/users/109691 | Automorphisms of the Lie algebras $\mathfrak{sl}(2,R)$ and $\mathfrak{su}(2)$ | It is clear that $PGL(2,\mathbb{R})$ acts as automorphisms. It is easy to check that the reflections act in a manner unlike any positive determinant matrices. Hence the automorphism group is larger than $PSL(2,\mathbb{R})$. Since we know the answer over $\mathbb{C}$ (as in Fulton and Harris, **Representation Theory**, ... | 13 | https://mathoverflow.net/users/13268 | 279474 | 123,828 |
https://mathoverflow.net/questions/279437 | 7 | Let $k$ be a field. Then consider the rings $k[T] / (T^n)$ with $n \in \mathbb{N}$. The inverse limit of these is given by $k[[T]]$. Passing on to the category of schemes, one concludes that the direct limit of $\text{Spec}(k[T] / (T^n)) $ is given by $\text{Spec}(k[[T]]) $.
Now to the question. $\text{Spec}(k[T] / (T^... | https://mathoverflow.net/users/113750 | Question concerning $\text{Spec}(k[[T]]) $ | I'm not sure if you''re looking for something deeper than this (and therefore would have preferred to make this a comment if I could have squeezed it in), but:
$Spec(A)$ has to be just rich enough so that every map from $A$ to a field shows up as a function on $Spec(A)$.
Now the issue is that, for a field $F$, the ... | 11 | https://mathoverflow.net/users/10503 | 279475 | 123,829 |
https://mathoverflow.net/questions/279301 | 1 | Does anyone know an example of an integral scheme $X$ over a field $k$ such that $X\_{\overline{k}}$ is connected but reducible? Does it make a difference if $k$ is perfect, or if we ask for $X\_{\overline{k}}$ to be reduced as well?
| https://mathoverflow.net/users/56878 | Example of an integral scheme which is geometrically connected but not geometrically irreducible | I am just posting my comment as an answer. For the field $\mathbb{R}$, the affine $\mathbb{R}$-scheme $\text{Spec}\ \mathbb{R}[x,y]/\langle x^2+y^2\rangle$ is integral and geometrically connected, but it is not geometrically irreducible. If $X\_k$ is an integral, locally finite type $k$-scheme that is *normal* and geom... | 5 | https://mathoverflow.net/users/13265 | 279476 | 123,830 |
https://mathoverflow.net/questions/279467 | 4 | Let $G=SL\_{2n}$ and let $\sigma:G \to G$ be defined by $\sigma (A)= E(A^t)^{-1}E^{-1}$, where $E=antidiag(1,1, ... ,1,-1,-1,...,-1)$. Then the maximal parabolic associated to the simple root $\epsilon\_n-\epsilon\_{n+1}$ is $\sigma$-stable, so $\sigma$ induces a map on $Gr(n,2n)$. The symplectic form associated to $E$... | https://mathoverflow.net/users/109750 | A map on Grassmannian | I am just posting my comment as an answer. I will change slightly your definition of $E$ so that the associated bilinear skew-symmetric form is the "standard" form, i.e., $$[ (x\_1,\dots,x\_{2n}), (y\_1,\dots,y\_{2n}) ] = (x\_1y\_{n+1}-x\_{n+1}y\_1) + \dots + (x\_ny\_{2n}-x\_{2n}y\_n).$$ For this symplectic form, the m... | 5 | https://mathoverflow.net/users/13265 | 279482 | 123,832 |
https://mathoverflow.net/questions/279440 | 2 | Given a probability space $\Omega$ and a countable set $M$ of measurable functions $f\colon \Omega\to \mathbb{R}$, I am looking for conditions on $M$ such that the following holds: For all $\varepsilon>0$, there exists a measurable partition $\Omega\_1,\ldots,\Omega\_n$ of $\Omega$, such that
$$
\sup\_{f\in M}\inf\_{g\... | https://mathoverflow.net/users/75786 | Approximate a set of functions by step functions on one partition | Your condition states precisely that your set is relatively compact in $L^2$. There are many other chracterisations---perhaps most famously the Kolmogorov one which is easy to find with google
| 0 | https://mathoverflow.net/users/113774 | 279485 | 123,834 |
https://mathoverflow.net/questions/279480 | 3 | I learned this interesting theorem but cannot remember from where since a long time has passed. What I'm wondering now is: Does someone know some constants which this theorem holds? ie, give some values of $a$, such that $[a^n]$ is **always prime** for $n\in N^+$(where $[\cdots]$ denotes integral part of a real number ... | https://mathoverflow.net/users/41499 | Constant $a$ such that $[a^n]$ is always prime for $n\in N^+$ | According to [Dubickas](http://www.mif.vu.lt/~dubickas/files/dvifai/monatshefte.pdf), the problem was still open in 2009; the conjecture is that no such $\alpha$ exists. Dubickas and his collaborators have worked extensively on integer parts of powers. See also [Baker&Harman](https://link.springer.com/article/10.1007%2... | 2 | https://mathoverflow.net/users/nan | 279486 | 123,835 |
https://mathoverflow.net/questions/279484 | 3 | Let $A$ be a $C^\*$-algebra, we denote with $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in matrices over $A$ (as usual). In <https://arxiv.org/pdf/math/0310340.pdf>, above definition 5.1. in section 5 (on page 20), there is mentioned the following fact:
If $A$ is a $C^\*$-algebra an... | https://mathoverflow.net/users/75338 | unital embedding into the coner $C^*$-algebra | Using your notation, we have diag$(x',...,x')\in M\_n\otimes M\_m(A).$ Then $M\_n$ is isomorphic to the algebra (let's call it $B$) generated by $e\_{ij}\otimes x'$ where $e\_{ij}$ are matrix units for $M\_n.$ Since $e\_{ij}\otimes x'$ (and hence everything in $B$) commutes with $vv^\*$, the map from $B$ to $pAp$ defin... | 3 | https://mathoverflow.net/users/34640 | 279501 | 123,839 |
https://mathoverflow.net/questions/279117 | 8 | Let $\mathcal{C}$ be a stable $\infty$-category. Let $Fun(\mathbb{Z},\mathcal{C})$ be the category of sequences of objects in $\mathcal{C}$. Where the category $\mathbb{Z}$ stands for the nerve of the poset $\mathbb{Z}$. There's a canonical functor:
$$Gr:Fun(\mathbb{Z},\mathcal{C}) \to
\underset{n \in \mathbb{Z}}{\c... | https://mathoverflow.net/users/22810 | Functorial construction of ("pre"-)spectral sequences? (Or - what is the "higher structure" underlying spectral sequences?) | Let me propose an answer to the question which isn't quite what you ask for. In fact, for the most part I agree with Denis that the correct object really is $\mathsf{Fil}(\mathcal{C})$. Also, I should mention that a really nice place to look for functorial discussions of spectral sequences is Verdier's thesis- most of ... | 7 | https://mathoverflow.net/users/6936 | 279510 | 123,840 |
https://mathoverflow.net/questions/278929 | 11 | I'm a big fan of [synthetic differential geometry](http://home.math.au.dk/kock/SGM-final.pdf) (or smooth infinitesimal analysis), as developed by Anders Kock and Bill Lawvere. It's a beautiful and intuitive geometric theory, which gives justification for the infinitesimal methods used by many of the pioneers of analysi... | https://mathoverflow.net/users/56938 | Relationship between synthetic differential geometry and differential cohesion? | I'm a co-author on the abstract linked in the comments but I'm coming from the computer science side so I'm not an expert on the models and I know very little classical differential geometry.
I had the same question and my current understanding is that
1. Differential Cohesion and Synthetic Differential Geometry h... | 7 | https://mathoverflow.net/users/82445 | 279524 | 123,842 |
https://mathoverflow.net/questions/279509 | 7 | **Question:** Given the long and skinny matrix $A\in\mathbb{R}^{m\times n}$ with $m\ge n$, define the matrix valued operator
$$\mathcal{A}:X\mapsto AX^{T}+XA^{T}.$$
What is the tightest nontrivial lower-bound on the singular value
$$\sigma\_{\min}(\mathcal{A})\triangleq\min\_{X}\{\|\mathcal{A}(X)\|\_{F}:\|X\|\_{F... | https://mathoverflow.net/users/60984 | Smallest singular value of $X\mapsto AX^{T}+XA^{T}$ | There is always a zero singular value as soon as $n \geq 2$.
Write $A = UD V$ with $D$ diagonal and $U, V$ orthogonal. Then we can write $X \mapsto AX^T + X A^T$ as $$X \mapsto UDV X^T + X V^T D^T U^T = U(D (U^T X V^T)^T + (U^T X V^T) D^T ) U^T$$ i.e. the composition of the operation $X \mapsto D X^T + X D^T$ with tw... | 5 | https://mathoverflow.net/users/18060 | 279526 | 123,843 |
https://mathoverflow.net/questions/279521 | -1 | C̶o̶n̶s̶i̶d̶e̶r̶ ̶a̶ ̶R̶i̶c̶a̶t̶t̶i̶ ̶e̶q̶u̶a̶t̶i̶o̶n̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶o̶r̶m̶
$$ y' + y^2 = S(x), \qquad \qquad \qquad (1)$$
w̶h̶e̶r̶e̶ ̶$̶S̶(̶x̶)̶$̶ ̶i̶s̶ ̶a̶ ̶m̶e̶r̶o̶m̶o̶r̶p̶h̶i̶c̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶,̶ ̶a̶n̶d̶ ̶$̶y̶$̶ ̶i̶s̶ ̶a̶ ̶c̶o̶m̶p̶l̶e̶x̶-̶v̶a̶l̶u̶e̶d̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶.̶ ̶D̶o̶e̶s̶ ̶t̶h̶e̶r̶e̶ ̶e̶x̶i̶s̶t̶ ̶... | https://mathoverflow.net/users/51685 | transforming a Ricatti equation into a generalised Ricatti equation | The transformation $y = -R(x) + s(x) u$ takes (1) to (2) with
$$\eqalign{P(x) &= {\frac { s \left( x \right)^{2}+s' \left( x \right) -S \left( x \right) }{R \left( x \right) }}
\cr
Q(x) &= -2 s(x)-\frac{R'(x)}{R(x)}}$$
| 2 | https://mathoverflow.net/users/13650 | 279528 | 123,844 |
https://mathoverflow.net/questions/179381 | 9 | My research area is mainly pro-$p$ groups and profinite groups. However, in the last few year I became also interested in discrete groups. Therefore, it seems to me a natural problem to look for examples of finitely generated groups such that their profinite completion is a pro-$p$ group. One trivial example is when th... | https://mathoverflow.net/users/5034 | When is the profinite completion a pro-$p$ group? | Gustavo A. Fernández-Alcober, Alejandra Garrido and Jone Uria-Albizuri gave a positive answer to question 2 and therefore also to question 1 in [On the congruence subgroup property for GGS-groups](http://www.ams.org/journals/proc/2017-145-08/S0002-9939-2017-13499-6/).
| 6 | https://mathoverflow.net/users/5034 | 279533 | 123,847 |
https://mathoverflow.net/questions/279303 | 6 | Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's [answer](https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658) to this MO [question](https://mathoverflow.net/questions/106560/philosophy-behin... | https://mathoverflow.net/users/85392 | Szpiro's conjecture for function fields and Mochizuki's approach to the number field case | You might want to also consider the geometric ("symplectic") version of the conjecture, since Mochizuki alredy has a paper outlining the relationship between Bogomolov's proof and his own IUT theory.
* Shinichi Mochizuki, "[Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of ... | 5 | https://mathoverflow.net/users/43108 | 279538 | 123,849 |
https://mathoverflow.net/questions/279127 | 14 | The probability a given integer in $[0,n]$ is a square is $\frac1{\sqrt n}$. What is the probability that if you take two integers uniformly then their product is square?
I know the main term is $\frac1n$. I am also looking for correction terms. The difficulty is an average integer has $\omega(\log\log n)$ factors w... | https://mathoverflow.net/users/10035 | Probability that product is a perfect square | For the case $m=k=2$, in which we seek the number $N(n)$ of pairs
$(x,y) \in [1,n]^2$ for which $xy$ is a square, we give
an elementary estimate
$$
N(n) = Cn \log n + An + O(n^{2/3}),
$$
where $C = 1/\zeta(2) = 6/\pi^2$ and
$$
A = \frac{3\gamma-1}{\zeta(2)}
- \frac{2\zeta'(2)}{\zeta(2)^2}
- 1
= 0.1377775\ldots \, .
$... | 11 | https://mathoverflow.net/users/14830 | 279541 | 123,852 |
https://mathoverflow.net/questions/279557 | 3 | Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma\_{\mathfrak{a}}(M)=\{x\in M\mid\mathfrak{a}\subseteq\sqrt{(0:\_Rx)}\}.$$ If $R$ or $M$ is noetherian, then this submodule has lots of nice properties... | https://mathoverflow.net/users/11025 | Torsion submodules of non-noetherian modules | Take the $k[x,y]$-module with generators $a\_n, n\in \mathbb N$ and relations
$$x a\_1= ya\_1=0$$ $$x a\_{2n}+ ya\_{2n+1} =a\_n$$
Let $\mathfrak a=(x,y)$, then clearly $\mathfrak a M = M$. The $\mathfrak a$-torsion submodule contains $a\_1$, so is nontrivial, and thus your second condition is clearly satisfies.
T... | 3 | https://mathoverflow.net/users/18060 | 279559 | 123,856 |
https://mathoverflow.net/questions/279490 | 7 | An F-space is a completely metrizable topological vector space, i.e. the vector topology is induced by a complete metric. A Fréchet space is, by definition, a locally convex F-space.
It is known that all (infinite dimensional) separable Fréchet spaces are homeomorphic to $l\_2$, the space of square summable sequences... | https://mathoverflow.net/users/27892 | Are separable F-spaces (completely metrizable topological vector space) homeomorphic to $l_2$? | There is a famous linear metric space constructed by R. Cauty [Un espace métrique linéaire qui n'est pas un rétracte absolu, Fund. Math. 146 (1994)] whose completion is a **separable $F$-space which is not an AR**. I do not have Cauty's paper handy but the latter fact is stated on the first page of [Cauty's space enh... | 6 | https://mathoverflow.net/users/1573 | 279562 | 123,859 |
https://mathoverflow.net/questions/279505 | 6 | Let $\phi\_1$ and $\phi\_2$ be the following statements:
$\phi\_1:$ There is a function $f:\{0,1\}^\*\to\{0,1\}$ computable in $E$ that has circuit complexity $2^{\Omega(n)}$.
$\phi\_2:$ There is a function $f:\{0,1\}^\*\to\{0,1\}$ computable in $NE \cap CoNE$ that has $2^{\Omega(n)}$ hardness on average.
>
> Q... | https://mathoverflow.net/users/83598 | Logical complexity of hard functions conjectures | As given, $\phi\_1$ and $\phi\_2$ are $\Sigma\_2$.
They *cannot* be shown equivalent to $\Pi\_2$ statements by any proof that *relativizes*. This follows by the same argument as in [Examples of $G\_\delta$ sets](https://mathoverflow.net/questions/57345/examples-of-g-delta-sets/57348#comment144034_57348) or <https://c... | 4 | https://mathoverflow.net/users/12705 | 279567 | 123,862 |
https://mathoverflow.net/questions/279143 | 2 | Let X be the union of two planes in $\mathbb{A}^4$ touching at origin. Blow up X at the origin. Call it $\overline{X}$. It has two disjoint copies of $\mathbb{A}^2$ blown up at the origins. Their exceptional divisors are $E\_1$ and $E\_2$ say. It is clear that $E\_1$ and $E\_2$ are isomorphic to $\mathbb{P}^1$. Choose ... | https://mathoverflow.net/users/nan | Gluing Schemes along subschemes | Let $Y$ be the total space of the vector bundle $O(-1)\oplus O(-1) \to \mathbb{P}^1$. Then your glued up scheme is isomorphic to the subscheme of $Y$ given by the union of the two closed subschemes given by the total spaces of the two line bundle $O(-1)\to \mathbb{P}^1$. So yes, your scheme is quasi-projective.
| 2 | https://mathoverflow.net/users/9617 | 279577 | 123,866 |
https://mathoverflow.net/questions/279397 | 7 | Let $\Sigma$ be an oriented, compact, connected 2-manifold with boundary. Assume that its boundary is equipped with a disjoint union decomposition into two non-empty parts:
$$\partial\Sigma=\partial\_{in}\Sigma\cup\partial\_{out}\Sigma$$
(both $\partial\_{in}\Sigma$ and $\partial\_{out}\Sigma$ are disjoint unions of ci... | https://mathoverflow.net/users/5690 | Handle decompositions using only 1-handles | The second statement ought to be in the literature somewhere but I don't know a reference so I'll give an argument.
The result can be rephrased in terms of graphs. Let $S$ be a compact connected surface with non-empty boundary and let $P$ be a non-empty finite set of points in the interior of $S$. Consider finite con... | 10 | https://mathoverflow.net/users/23571 | 279579 | 123,867 |
https://mathoverflow.net/questions/279552 | 9 | I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.
When I approached the speaker after the talk whether he could state in a general way what this concept says he was very resilient to state something th... | https://mathoverflow.net/users/112877 | Concentration compactness. Can this concept be stated in a theorem? | This is really just a longwinded comment.
The earliest instances of concentrated compactness that I know of are for geometric questions such as existence of energy-minimizing harmonic maps (as studied by Sacks and Uhlenbeck), the Yamabe problem (as studied by Trudinger, Aubin, and Schoen), and the existence of self-d... | 3 | https://mathoverflow.net/users/613 | 279589 | 123,871 |
https://mathoverflow.net/questions/268362 | 13 | $\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\sAverage}[1]{\langle#1\rangle} $
$\newcommand{\IP}[2]{\sAverage{#1,#2}}$
$\newcommand{\C... | https://mathoverflow.net/users/46290 | Is this expression for the Laplacian of conformal maps between Riemannian manifolds known? | Well, *for conformal maps* equation $(1)$ is merely $d$-harmonicity in disguise:)
The equation is
$$
\delta\big((\det df)^{1-\frac{2}{d}} df\big)=0. \tag{1}
$$
Since for conformal maps, $\det df=\|df\|^d$ up to a constant, we equivalently get
$$
\delta\big(\|df\|^{d-2} df\big)=0. \tag{2}
$$
(Which is trivial of... | 1 | https://mathoverflow.net/users/46290 | 279590 | 123,872 |
https://mathoverflow.net/questions/279458 | 10 | **Notations:** Let $L\_\alpha$ stand for the Gödel constructible hierarchy ($L\_0=\varnothing$ and $L\_{\alpha+1} = \mathrm{def}(L\_\alpha)$ is the set of definable subsets of $L\_\alpha$ and $L\_\delta = \bigcup\_{\beta<\delta} L\_\beta$ for limit $\delta$), and $J\_\alpha$ for the Jensen hierarchy ($J\_0=\varnothing$... | https://mathoverflow.net/users/17064 | Stability for the Gödel and Jensen hierarchies | As remarked the non-trivial direction is to show $L\_\sigma\prec\_{\Sigma\_1}L\_\gamma$ implies $J\_\sigma\prec\_{\sigma\_1}J\_\gamma$. Let's take the extreme case that $\gamma=\sigma+1$. Suppose $J\_{\sigma+1}\models \exists u \varphi(u,x)$ where $\varphi\in\Sigma\_0$ and $x\in J\_\sigma = L\_\sigma$. The short answer... | 11 | https://mathoverflow.net/users/6942 | 279593 | 123,873 |
https://mathoverflow.net/questions/279595 | 5 | Let $Q\_1, Q\_2, R$ be quadratic froms over $\mathbb{Z}$ such that $Q\_1 \oplus R \cong Q\_2 \oplus R$ as quadratic forms. Is it necessary that $Q\_1 \cong Q\_2$?
I know that by Witt's theorem it is true for fields.
| https://mathoverflow.net/users/49822 | Are stably equivalent quadratic forms over Z equivalent? | I would say no; an example is when $R$ is a hyperbolic plane, the relation you demand says merely that $Q\_1$ and $Q\_2$ are in the same genus. This is in SPLAG, page 378 in the first (1988) edition, see also [Clark Jagy](http://alpha.math.uga.edu/~pete/Clark_Jagy_11_13_2013.pdf).
Notice that rational equivalence "wi... | 8 | https://mathoverflow.net/users/3324 | 279596 | 123,875 |
https://mathoverflow.net/questions/279123 | 8 | Note: This question now has a [sister](https://mathoverflow.net/questions/279618/what-is-the-relation-between-coxeter-transformations-of-coxeter-systems-and-coxe) :-)
**The Coxeter transformation of a generalized Cartan matrix**:
In the paper
[The spectral radius of the Coxeter transformations for a generalized Car... | https://mathoverflow.net/users/57296 | What is the relation between Coxeter transformations of generalized Cartan matrices and Coxeter transformations of finite-dimensional algebras? | Since I didn't want to think about permutations too much, here is an answer about relating Coxeter transformations of the form $C(A, \text{id})$ to Coxeter transformations of finite-dimensional path-algebras.
Let $A$ be a generalized Cartan matrix. Let $A\_+$ and $A\_-$ be defined by
$$\left( A\_+\right)\_{ij} =
\b... | 5 | https://mathoverflow.net/users/57296 | 279598 | 123,876 |
https://mathoverflow.net/questions/279591 | 1 | For any positive integer $n\in\mathbb{N}$ let $S\_n$ denote the set of all permutations (bijections) $\pi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. For any $\pi\in S\_n$ we let the *maximal displacement* be defined by $$\text{maxd}(\pi)= \max\big\{|k - \pi(k)|: k\in \{1,\ldots,n\}\big\}.$$
The expected value of the maximal di... | https://mathoverflow.net/users/8628 | Expected value of maximal displacement in permutations of $\{1,\ldots,n\}$ | Fleshing out Boris Bukh's idea.
We can draw $\pi$ by first sending $1$ uniformly to somewhere in $\{1,\dots,n\}$, then sending $2$ uniformly to the remaining $n-1$ spots, and so on.
Consider a small but superconstant $m$, in particular $m = n^{2/3}$ works. The idea is that one of the first $m$ elements almost certa... | 3 | https://mathoverflow.net/users/29697 | 279599 | 123,877 |
https://mathoverflow.net/questions/279148 | 5 | I am not sure that this is a research level question.
Remark 10.9.4 in the book "A course in metric geometry" by Burago, Burago, Ivanov claims the following.
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below. Fix $p\in X$ and $\varepsilon> 0$. Then there exists $r>0$ such that for any two... | https://mathoverflow.net/users/16183 | Angle estimate in Alexandrov spaces | Partly following the advise in Anton Petrunin's answer, let me present a proof in greater detail.
We prove the statement in the case of non-negative curvature. Assume the contrary. Then there exist $\delta >0$ and sequences $x\_n,y\_n\to p$ such that $0<|px\_n|\leq |py\_n|\to 0$, and
\begin{eqnarray}\label{0}
\tilde... | 2 | https://mathoverflow.net/users/16183 | 279611 | 123,881 |
https://mathoverflow.net/questions/279573 | 5 | I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints:
1. $X$ has the same Betti numbers as $\mathbb{C}\mathbb{P}^{3}$ i.e. $b\_{1}(X) = b\_{3}(X) = 0$ and $b\_{2}(X) = 1$ and all of its cohomology groups are torsion-free.
2.... | https://mathoverflow.net/users/99732 | 3-folds with "simple" Betti numbers and positive Kodaira dimension | Let me just mention that the non-existence of such a threefold is an immediate consequence of Yau's inequality. First, as explained in the above comment, the conditions $b\_2=1$ and $\mathrm{Kod}(X)\geq 0$ imply that $K\_X$ is ample. Then Yau gives $c\_1^3\geq \frac{8}{3}c\_1c\_2 $, which is equivalent by Riemann-Roch ... | 8 | https://mathoverflow.net/users/40297 | 279615 | 123,882 |
https://mathoverflow.net/questions/279620 | 8 | Let $V$ be a finite-dimensional vector space over a field $K$.
Let $U$ be a linear subspace of $\mathrm{End}(V)$. Write $UV$ for the span of all $Av$ where $A\in U$ and $v\in V$. Suppose that
$$
\ker(U)=\bigcap\_{A\in U}\ker(A)
$$
is zero and that
$$
\mathrm{coker}(U)=V/UV
$$ is zero.
Is it true that $U$ must contain a... | https://mathoverflow.net/users/nan | When does a space of endomorphisms contain invertibles? | Nope:
$$
\left\{
\begin{bmatrix}
0 & x & y\\
x & 0 & 0\\
y & 0 & 0
\end{bmatrix}: x,y \in\mathbb{R}
\right\}.
$$
This is a classical counterexample in numerical linear algebra -- the simplest singular matrix pencil with a nontrivial Kronecker canonical form.
| 9 | https://mathoverflow.net/users/1898 | 279624 | 123,888 |
https://mathoverflow.net/questions/279487 | 4 | How to derive an upper bound on the minimum number $n(k,d)$ of lattice points in $d$-dimensions such that there are some $k$ of these points which have a lattice point centroid.
| https://mathoverflow.net/users/113775 | Upper Bound on minimum number of lattice points | The trivial inequality $$n(k, d) \leq k \times n(k, d - 1) - (k - 1)$$ gives a trivial upper bound: $$n(k, d) \leq (k- 1) k^d + 1.$$
---
An easy lower bound is: $$n(k, d) \geq (k - 1) 2^d + 1.$$ This is proved by choosing the following lattice points: for every vector in $\{0, 1\}^d \subseteq (\mathbb{Z}/k\mathbb... | 2 | https://mathoverflow.net/users/76332 | 279626 | 123,889 |
https://mathoverflow.net/questions/279625 | 3 | Let $X$ be the following vector field on the plane:
$$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$
The vector field $ (X)$ has a non isochronous center at the origin.The proof is given in Remark $2$ below. The punctured plane is filled with periodic orbits of $X$. The vector field is geodesible on the p... | https://mathoverflow.net/users/36688 | An explicit formula for a flat metric compatible to certain polynomial vector field with center | Since the metric doesn't have to extend to the origin, take the flat metric
$$
g = \frac{\bigl(\mathrm{d}\left(x\sqrt{1+x^2/2}\right)\bigr)^2 + \mathrm{d}y^2}{x^2+x^4/2+y^2}.
$$
The level curves $x^2+x^4/2+y^2 = r^2$ are geodesics for the metric, and these are the integral curves of the vector field $X$.
| 8 | https://mathoverflow.net/users/13972 | 279628 | 123,890 |
https://mathoverflow.net/questions/279629 | 10 | It is a theorem of Alperin, Feit and Thompson [Isaacs Character Theory book (4.9)] that if $G$ is a $2$-group in which the number of involutions is congruent to $1$ modulo $4$, then $G$ is cyclic or $|G:G'|=4$. In the latter case $G$ is dihedral, semidihedral or generalized quaternion.
Are there any results of a simi... | https://mathoverflow.net/users/20764 | Number of involutions in a finite group modulo $4$ | EDIT: Actually, there is an older reference which will give the result below. See
>
> Herzog, Marcel
> Counting group elements of order $p$ modulo $p^2$.
> Proc. Amer. Math. Soc. 66 (1977), no. 2, 247–250.
>
>
>
where you will also find references to other related papers.
------
------
The following pape... | 9 | https://mathoverflow.net/users/10146 | 279634 | 123,892 |
https://mathoverflow.net/questions/279399 | 15 | Let $T$ be a ergodic automorphism of a non-atomic Lebesgue probability space $(X, \mathcal{A}, \mu)$.
The celebrated Rokhlin tower lemma says that given an integer $n>0$ and $0 < \epsilon < 1$, there exists $B \in \mathcal{A}$ such that the sets $B$, $TB$, ..., $T^{n-1} B$ are disjoint and their union (called a *towe... | https://mathoverflow.net/users/1516 | A Rokhlin lemma with a prescribed height function? | So I've changed my mind! I think the answer is "yes". Suppose $X$ and $N$ are given. Let $\epsilon>0$ be given and let $M$ be such that $\mu({x:N(x)>M})<\epsilon$. Now build a Rokhlin tower with height $M/\epsilon$ and error set of size at most $\epsilon/M$. Let $A$ denote the base of the tower (so that $\mu(A)<\epsilo... | 5 | https://mathoverflow.net/users/11054 | 279635 | 123,893 |
https://mathoverflow.net/questions/279639 | 3 | Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that
$$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did not come to any good result, but I think the solution should be difficult.
| https://mathoverflow.net/users/113840 | Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? | No, the constant functions $$f=\frac{-1\pm\sqrt 5}{2}$$ are both solutions.
| 2 | https://mathoverflow.net/users/15517 | 279640 | 123,894 |
https://mathoverflow.net/questions/279636 | 4 | Let $X$ be a projective variety over a field $K$ of characteristic zero. Denote by $p:X\_{\overline{K}} \to X$ the natural morphism, where $\overline{K}$ is the algebraic closure of $K$ and $X\_{\overline{K}}:=X \times\_K \overline{K}$. Let $E$ and $F$ be coherent sheaves on $E$ such that $p^\*E \cong p^\*F$. Does it i... | https://mathoverflow.net/users/58203 | Projection formula for field extension | The answer is affirmative with $X$ any proper scheme over any field $K$, moreover using any field extension $K'/K$ in place of $\overline{K}/K$.
Let $H\_{E,F} = \mathscr{H}om(E,F)$, a coherent sheaf on $X$, and define $H\_{F,E}$ and $H\_{E,E}$ similarly. Thus, $H\_{E,F}(X) = {\rm{Hom}}\_{O\_X}(E,F)$ is a finite-dime... | 8 | https://mathoverflow.net/users/81332 | 279643 | 123,896 |
https://mathoverflow.net/questions/279614 | 8 | When searching for the originator of the method of removing radicals from equations I found the following remark by [D. Mooney](https://books.google.de/books?id=rpU_AAAAYAAJ&pg=PA221&redir_esc=y#v=onepage&q&f=false):
>
> *"In the 97th Section of his Analysis, Doctor Hales shews the method of taking quadratic surds... | https://mathoverflow.net/users/31310 | Fermat's Method of Removing Radicals | Fermat developed his method of clearing radicals to solve the problem of finding the maximum and minimum of a polynomial, as an application of differential calculus "avant la lettre". A 19th century [source](https://books.google.nl/books?id=PttEAQAAMAAJ&pg=PA441&lpg=PA441&source=bl&ots=huPbrRC_Wq&sig=UewkhU_hFqCGN9gpwL... | 3 | https://mathoverflow.net/users/11260 | 279650 | 123,897 |
https://mathoverflow.net/questions/279656 | 52 | **Question 1**
Is there a winning strategy (algorithm to play infinitely) in Tetris,
or is there a sequence of bricks which is impossible to pack without holes?
Consider generalized Tetris with Young diagrams (for some $n$) are falling down.
**Question 2** Is there winning strategy?
If not - consider some probabil... | https://mathoverflow.net/users/10446 | Is there winning strategy in Tetris ? What if Young diagrams are falling? | Heidi Burgiel's first paper "How to Lose at Tetris" (which she wrote towards the end of our time in grad school in Seattle) answers Question 1 in the negative. It was published in the *Mathematical Gazette* volume 81 (1997) 194--200. If you want to see the published version, that volume is still on JSTOR where you can ... | 48 | https://mathoverflow.net/users/14807 | 279662 | 123,903 |
https://mathoverflow.net/questions/279618 | 2 | This question is related to the following [question about Coxeter transformations](https://mathoverflow.net/questions/279123/what-is-the-relation-between-coxeter-transformations-of-generalized-cartan-matri) that I asked and recently answered myself. For completeness I also write full definitions in the new question.
... | https://mathoverflow.net/users/57296 | What is the relation between Coxeter transformations of Coxeter systems and Coxeter transformations of generalized Cartan matrices? | The point of your question is: What is the relation between the two representations you describe of a Coxeter group?
To each Coxeter group, one can associate a Cartan matrix in an even more general sense, where you allow real entries but require that $a\_{ij}a\_{ji}=4\cos^2\left(\frac{\pi}{m(i,j)}\right)$, where the... | 3 | https://mathoverflow.net/users/5519 | 279665 | 123,905 |
https://mathoverflow.net/questions/279644 | 4 | Let $G$ be a reductive algebraic group over $\mathbb{C}$. Let $\operatorname{Gr}\_{G}$ be the corresponding affine Grassmannian ($\operatorname{Gr}\_{G}(\mathbb{C})=G(\mathbb{C}((z)))/G(\mathbb{C}[[z]])$). Let $\operatorname{Perv}\_{G(\mathbb{C}[[z]])}\operatorname{Gr}\_{G}$ be the category of $G(\mathbb{C}[[z]])$-equi... | https://mathoverflow.net/users/113438 | Computation of multiplicity of irreducible representation in some representation via geometric Satake correspondence | Yes, the multiplicity space is given by $\mathcal H\_{z^\lambda}^{- dim Gr^\lambda}(\mathcal P) $. See, for example, the argument given in the proof of Proposition 3.1 of <https://arxiv.org/pdf/math/0304176.pdf>.
| 2 | https://mathoverflow.net/users/438 | 279668 | 123,907 |
https://mathoverflow.net/questions/279659 | 7 | Suppose $P$ and $Q$ are ccc partial orders. Is $P \times Q$ $\omega\_2$-cc? Note that this true under CH by the Erdos-Rado Theorem.
| https://mathoverflow.net/users/11145 | chain condition of a product of posets | First of all, note (as Monroe does in his question) that if $\mathbb P,\mathbb Q$ are ccc, then $\mathbb P\times\mathbb Q$ is $\mathfrak c^+$-cc, as an immediate consequence of the Erdős-Rado theorem $(2^{\aleph\_0})^+\to(\aleph\_1)^2\_2$. (This is to say, if $\mathbb P$ and $\mathbb Q$ do not admit uncountable anticha... | 10 | https://mathoverflow.net/users/6085 | 279680 | 123,910 |
https://mathoverflow.net/questions/279685 | 1 | In Barwise's book, *Admissible Sets and Structures*, the following statement is made (on pg. 8):
>
> "...if $ZF$ is consistent, so is $ZF$ + "There is no transitive model of $ZF$"
>
>
>
He mentions this fact as an example of how $ZF$ is, in some ways, "too strong" (also found on pg. 8):
>
> (1) The most ob... | https://mathoverflow.net/users/20597 | Is there a 'Constructible Universe' that is a submodel of a non-transitive model of $ZF$? | What $\sf ZF$ actually proves is that there is a class called $L$, and for every axiom of $\sf ZFC$, the relativization of that axiom holds in $L$.
Therefore, the fact we are talking about non-transitive models does not matter. They have their own version of $L$ anyway.
What is provable is that $L$ is a transitive... | 8 | https://mathoverflow.net/users/7206 | 279687 | 123,912 |
https://mathoverflow.net/questions/279627 | 0 | Let $A$ be a finite dimensional symmetric algebra over a field (we can also assume that it is connected).
Call a non-projective indecomposable module $M$ strange in case $Ext^i(M,M)=0$ for all but finitely many $i$.
Is it true that strange $M$ have complexity equal to one, which means that the terms in a minimal projec... | https://mathoverflow.net/users/61949 | Strange modules part II | Let $M$ be one of the examples you know. Then $M\otimes\_kM$ is a strange module for $A\otimes\_kA$, but does not have complexity one.
| 1 | https://mathoverflow.net/users/22989 | 279688 | 123,913 |
https://mathoverflow.net/questions/279682 | 7 | Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form of the boundary is at least $\lambda$.
**Does there exist an upper bound on the area of $X$ in terms of $\kappa, D,\lam... | https://mathoverflow.net/users/16183 | Estimate of area of 2-dimensional surface | Yes, there is a bound; your problem can be reduced to the case of convex boundary using the following trick.
Without loss of generality, we can assume that $\lambda=-\tfrac1{10}$ and $\kappa=-1$.
In this case you can attach a collar to your surface locally isometric to the tubular neighborhood in of line in the Lob... | 6 | https://mathoverflow.net/users/1441 | 279698 | 123,919 |
https://mathoverflow.net/questions/279515 | 1 | Let $R$ be an integral domain. Consider the set $$S := \big\{a \in R\smallsetminus \{0\}
: Ra+Rx \text{ is a principal ideal } \forall x \in R \big \}.$$ Is $S$ a saturated multiplicative closed subset of $R$? If in general $S$ is not saturated or multiplicative closed, what if we assume $R$ is a GCD domain? Is the c... | https://mathoverflow.net/users/nan | On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind | The answer is **yes** if $R$ is any [atomic domain](https://en.wikipedia.org/wiki/Atomic_domain), e.g., $R$ is a Noetherian domain.
>
>
> >
> > **Claim 1.** Let $R$ be any integral domain. The set $S = S\_R$ is saturated in the sense that if $ab \in S\_R$ for some $a,b \in R$, then $a \in S\_R$.
> >
> >
> > *Pr... | 5 | https://mathoverflow.net/users/84349 | 279713 | 123,924 |
https://mathoverflow.net/questions/279711 | 7 | Suppose I've got vectors $v = (1,-1)$ and $w = (1,1)$ and any $m \in \mathbb{N}$. Let $a = v \otimes v \otimes w^{\otimes m}$ and let $\tilde{a}$ be the sum over all $\binom{m}{2}$ unique vectors obtained by permuting the tensor coordinates of $a$. I'm interested in identifying the asymptotics of a function $f(m) = \|\... | https://mathoverflow.net/users/86053 | Counting with tensor products | You are looking at the vector of coefficients of the polynomial
$$\sum\_{\epsilon}\prod\_{i=1}^{m+2}(1-\epsilon\_i x\_i)$$
where $\epsilon$ runs over all choices of signs $\pm$ provided there are exactly two $-$ signs. There are $\binom{m+2}{2}$ terms being summed. After expanding we get a multilinear symmetric polynom... | 7 | https://mathoverflow.net/users/2384 | 279716 | 123,925 |
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