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https://mathoverflow.net/questions/250726 | -2 | I recently came across the following statement,
The Galois coverings of $\mathbb{P}^1\_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1\_L\to\mathbb{P}^1\_\mathbb{Q}$$ where $L$ is a number field.
How to prove this ,
Any reference.
| https://mathoverflow.net/users/92070 | Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$ | Let $X$ be a smooth curve over a field $K$. Then there is an exact sequence of etale fundamental groups:
$$1\rightarrow\pi\_1(X\_{\overline{K}})\rightarrow\pi\_1(X)\rightarrow\pi\_1(Spec\;K)\rightarrow 1$$
Taking $X= \mathbb{P}^1\_K$, we find that $X\_{\overline{K}}$ has trivial etale fundamental group (if $K$ is a... | 3 | https://mathoverflow.net/users/15242 | 250727 | 113,828 |
https://mathoverflow.net/questions/250643 | 3 | Let $H$ be a Hopf algebra and $K$ a Hopf subalgebra of $H$. If $H$ is finite-dimensional, then by the Nichols-Zoeller Theorem $H$ is free as a left (and right) module over $K$.
Moreover, the same conclusion holds when $H$ is pointed, by the main theorem of
[D. Radford: Pointed Hopf algebras are free over Hopf suba... | https://mathoverflow.net/users/17582 | Bases of free Hopf algebras over Hopf subalgebras | In the pointed case, this should follow from the proof in the Radford's paper available at the quoted link.
In your notation, consider the left $K$-module $C=KG$, where $G$ is the group of group-like elements of $H$. It is easy to see (and proved in the paper) that a basis of $C$ over $K$ is given by any set of right... | 3 | https://mathoverflow.net/users/14653 | 250740 | 113,831 |
https://mathoverflow.net/questions/250720 | 4 | Let $\mathcal{M}\_{0,n}$ denote the moduli stack of $\mathbb{P}^1$'s equipped with $n$ distinct sections. My understanding is that $\mathcal{M}\_{0,3}$ is a point, so $\mathcal{M}\_{0,4}$ is a scheme, isomorphic to $\mathbb{P}^1 - \{0,1,\infty\}$. This has fundamental group $F\_2$ (the free group of rank 2), and so $\m... | https://mathoverflow.net/users/88840 | Fundamental groups of $\mathcal{M}_{0,n}$ | Over an algebraically closed field of characteristic zero, a choice of complex analytification gives you an equivalence between finite étale covers of the moduli scheme $\mathcal{M}\_{0,n}$ and finite covers of the configuration space of $n$ ordered points on the 2-sphere.
We can compute the fundamental group of the ... | 7 | https://mathoverflow.net/users/121 | 250741 | 113,832 |
https://mathoverflow.net/questions/250735 | 2 | Consider a domain $\mathcal{D}$ which is a right circular cylinder in $\mathbb{R}^3$, with radius 1 and height 1, say. The boundary of $\mathcal{D}$, which I denote by $\partial\mathcal{D}$ consists of two parts:
(a) The unit disk on the plane $z=0$, which I denote by $\mathcal{F}$.
(b) The unit disk on the plane ... | https://mathoverflow.net/users/98946 | Showing existence of minimisers with single integral constraint on a possibly non-Lipschitz domain? | Your domain **is** a Lipschitz (graph) domain. The 3-dimensional (!) Lebesgue measure of $\mathcal{F}$ is indeed zero, whereas $\mathcal{F}$ has positive measure for the boundary measure $\sigma$ (which coincides with the 2-dimensional Hausdorff measure). This should help you to solve your other questions. The second a... | 1 | https://mathoverflow.net/users/85906 | 250743 | 113,833 |
https://mathoverflow.net/questions/248600 | 13 | $\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}\_1}
\newcommand{\spec}{\operatorname{Spec}}$If I understand correctly, in Borger's paper [$\Lambda$-rings and the field with one element](https://maths-people.anu.edu.au/%7Eborger/preprints/01/lrfoe13.pdf) about the field with one element, the category of "affine ... | https://mathoverflow.net/users/50409 | Is an ordinary scheme in Borger's Absolute Geometry the same as a "scheme over ₁" with a map to Spec(ℤ)? | I guess I should be the one to answer this!
Unfortunately, the answer is no.
There is a natural adjunction between the two types of data: given a $\Lambda$-ring $R$, set $R\_0$ to be $R\otimes\_{W(\mathbb{Z})}\mathbb{Z}$, where the map $W(\mathbb{Z})\to\mathbb{Z}$ is the projection on the first component, i.e. the ... | 16 | https://mathoverflow.net/users/1114 | 250744 | 113,834 |
https://mathoverflow.net/questions/250755 | -3 | Let $E$ be an elliptic curve over $\mathbb{Q}.$
Is $\sharp E((\mathbb{F}\_{p^{2}})/E(\mathbb{F}\_{p}))=1$ for almost all primes $p$?
| https://mathoverflow.net/users/70751 | Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$? | No, this is almost never true. $|\# E(\mathbb{F}\_q) -q - 1| \leq 2 \sqrt{q}$. So $\#E(\mathbb{F}\_{p^2}) \geq p^2 - 2p +1=(p-1)^2$ and $\#E(\mathbb{F}\_p) \leq p + 2 \sqrt{p} +1= (\sqrt{p}+1)^2$. So $\#E(\mathbb{F}\_{p^2})/\#E(\mathbb{F}\_p) \geq (\sqrt{p}-1)^2$. If $p \geq 5$, this is $>1$.
Let's look a little clos... | 7 | https://mathoverflow.net/users/297 | 250758 | 113,837 |
https://mathoverflow.net/questions/250161 | 1 | I am trying to understand the motivation behind the main theorem of Keraani: See the top of [page 356](http://www.sciencedirect.com/science/article/pii/S0022039600939512) and which is the motivation behind his Theorem 1.6
---
We consider
$$i\partial\_t u + \Delta u =0, u(x,0)=u\_0 \in H^{1}(\mathbb R^3)$$
where ... | https://mathoverflow.net/users/96950 | Defect of Compactness for the Strichartz Estimates | You have two spaces $X$ and $Y$. You have a linear operator $L: X\to Y$ that is bounded (continuous). (Here $X = H^1(\mathbb{R}^d)$ and $Y$ is the Strichartz norm space on $\mathbb{R}^{d+1}$. The linear operator is $e^{it\Delta}$, the solution operator to the free Schrodinger equation.) So when the author says that the... | 2 | https://mathoverflow.net/users/3948 | 250760 | 113,838 |
https://mathoverflow.net/questions/250737 | 4 | Let $q$ be an odd number...
consider $0-1$ strings of length $2q$ with $q$ ones. [with total number of $C(2q,q)$]
I want to find an upper bound for a set of these strings such that the number of mutual ones in any two of them be an odd number...
---
i already wrote a code in MATLAB and it seems this number i... | https://mathoverflow.net/users/92217 | Size of biggest mutually 0-1 string with odd mutual 1 | The answer is $2^{q+o(q)}$ (well, there are rooms for improvement).
Example: consider all strings with $1$ at the last position and $(q-1)/2$ pairs of one's chosen from possible pairs of positions $(1,2)$, $(3,4),\dots$, $(2q-3,2q-2)$. It is a family consisting of $\binom{q-1}{(q-1)/2}$ strings.
For the estimate, ... | 6 | https://mathoverflow.net/users/4312 | 250764 | 113,839 |
https://mathoverflow.net/questions/250763 | 0 | Is the hyperbolic version of [Sylvester co linear problem](https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem) true?
| https://mathoverflow.net/users/36688 | Hyperbolic version of Sylvester co-linear problem | Yes. Use the projective (Cayley-Klein) model of the hyperbolic plane. Your points lie inside a disk. Hyperbolic lines are chords of that disk. Since the Euclidean Silvester-Gallai is true, you have the same conclusion in the hyperbolic case.
| 6 | https://mathoverflow.net/users/98590 | 250765 | 113,840 |
https://mathoverflow.net/questions/250773 | 1 | If $X\_n \sim N(\mu,\sigma)$ and $T\_n = \frac{1}{n}\sum\_1^n X\_i$
What is the rate of convergence of $e^{T\_n}$ to $e^{\mu}$
| https://mathoverflow.net/users/nan | Rate of convergence of exponential of sample mean to exponential of first moment? | $T\_n \sim N(\mu, \sigma/\sqrt{n})$, and $e^{T\_n}$ has a lognormal distribution.
Its mean and variance are $$\eqalign{\exp(\mu &+ \sigma^2/(2n))\cr
&= \exp(\mu) \left(1 + \dfrac{\sigma^2}{2n} + O\left(\frac{1}{n^2}\right)\right)}$$ and $$\eqalign{\exp(2\mu &+ 2\sigma^2/n) - \exp(2\mu + \sigma^2/n)\cr &= \exp(2\mu) \df... | 1 | https://mathoverflow.net/users/13650 | 250776 | 113,842 |
https://mathoverflow.net/questions/250772 | 3 | Assume that we have $\epsilon\_1, \; \epsilon\_2$ independent white noises.
1. Can I write $\int\_{0}^1 \epsilon\_1^2(t)dt$
2. Can I write $\int\_{0}^1 \epsilon\_1(t) \epsilon\_2(t)dt$
1 and 2 obviously make no sense in $L^2$ nor in terms of Wiener integral. Is there any way I can make sense out of it?
| https://mathoverflow.net/users/98395 | can I integrate product or square of a white noise in any sense? | A consistent framework for "nonlinear stochastic calculus" has been developed in [The square of white noise as a Jacobi field](https://arxiv.org/abs/math/0401370v1) (2004), building on earlier work in [Squared white noise and other non-Gaussian noises as Lévy processes on real Lie algebras](https://art.torvergata.it/re... | 3 | https://mathoverflow.net/users/11260 | 250779 | 113,845 |
https://mathoverflow.net/questions/250748 | -1 | Let $A$ be a k-algebra,where k is a fixed field. We denote by $\mathfrak{D}^b(A-mod)$ the bounded derived A-module category. A complex $Z^{\bullet}=(Z^i,d^i) \in \mathfrak{D}^b(A-mod)$ such that all $Z^i$ are finitely generated projective is quasi-isomorphic to the following complex $$ \cdots \rightarrow 0 \rightarrow ... | https://mathoverflow.net/users/83554 | How to show the following properties of $Coker(d^{-n-1})$? | Well, how do we conclude that our complex $Z^\bullet$ is quasi-isomorfic to the complex
$$
\cdots \longrightarrow 0 \longrightarrow Coker(d^{-n-1}) \longrightarrow Z^{-n+1} \longrightarrow \cdots \longrightarrow Z^{m-1} \longrightarrow Ker(d^m) \longrightarrow 0 \longrightarrow \cdots \ ?
$$
We use the two truncat... | 1 | https://mathoverflow.net/users/95546 | 250790 | 113,848 |
https://mathoverflow.net/questions/249919 | 19 | The group $S\_{24}$ of permutations of $24$ things has fourth integral cohomology $\mathrm H^4(S\_{24};\mathbb Z) \cong \mathbb Z/2 \oplus \mathbb Z/2 \oplus \mathbb Z/12$. According to [Sikiric and Ellis](https://arxiv.org/abs/0812.4291) the [largest Mathieu group](https://en.wikipedia.org/wiki/Mathieu_group_M24) $M\_... | https://mathoverflow.net/users/78 | Is the map $\mathrm H^4(S_{24}) \to \mathrm H^4(M_{24})$ surjective? | The answer to my question is No. The generator of the $\mathbb Z/12$ part of $H^4(S\_{24})$ is $p\_1$ of the permutation representation. That representation restricts to $M\_{24}$ to a Spin representation, i.e. one with $w\_1=w\_2=0$. For any such representation, $p\_1$ is automatically even.
Thus the map $H^4(S\_{24... | 11 | https://mathoverflow.net/users/78 | 250792 | 113,850 |
https://mathoverflow.net/questions/250767 | 4 | I have a robot which moves abiding by the following equation:
$$
\frac{d^2 \mathbf{x}}{dt} = -k \frac{d\mathbf{x}}{dt} + \mathbf{a}(t),
$$
where $\mathbf{x}$ is a coordinate vector.
It starts with
$$
\frac{d\mathbf{x}}{dt} = v\_0,$$
and $\mathbf{x} = (x0, y0, z0)$.
I need to find continuous vector valued acceleration... | https://mathoverflow.net/users/49122 | Optimal control of a robot | Then if the constraint concerns the $L^\infty$ norm, that is, $\forall i \in \{x, y, z\} \leq c$, in each individual direction you have a probelm of a starting velocity, a target endpoint, and a maximum acceleration. You easily solve (for each coordinate) for the time taken to get to that target using the maximal accel... | 4 | https://mathoverflow.net/users/82067 | 250795 | 113,852 |
https://mathoverflow.net/questions/250797 | 12 | For which $n$ is the "principal congruence subgroup" $\Gamma(n)\le \mathrm{SL}\_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic? I.e., for which $n$ is $\Gamma(n)$ stable (as a set) under all automorphisms of $\mathrm{SL}\_2(\mathbb{Z})$?
| https://mathoverflow.net/users/88840 | are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$? | No for $n$ odd, yes for $n$ even.
By a result of [Hua and Reiner](http://www.ams.org/journals/tran/1951-071-03/S0002-9947-1951-0043847-X/home.html), the automorphism group of $SL\_2(\mathbb{Z})$ is generated by (1) conjugation by $GL\_2(\mathbb{Z})$ (the matrices with determinant $\pm 1$) and (2) the map $X \mapsto \... | 19 | https://mathoverflow.net/users/297 | 250799 | 113,853 |
https://mathoverflow.net/questions/250745 | 1 | Let $(L\_t)\_{0\leq t \leq 1}$ be the local time at $0$ of a brownian bridge. Let $(T\_l)\_{0\leq l \leq L\_1}$ be its generalized inverse (as in $T\_l := \inf\{t\geq 0 : L\_t \geq l\}$). What is the joint distribution of
$$(T\_{L\_1/4}, T\_{L\_1/2}, T\_{3L\_1/4}) ?$$
In particular, I am interested in the joint distr... | https://mathoverflow.net/users/98954 | Quartiles of local time of brownian bridge at origin | *Joint Law of Brownian Motion and Its Local Time*.
In what follows, we will use the joint law of Brownian motion $B\_t$ with $B\_0=x \in \mathbb{R}$ and its local time accumulated at zero $\ell\_t$, which can be found, e.g., in (1.3.8) on pg. 140 of:
A. Borodin and P. Salminen (1996). **Handbook of Brownian Motion*... | 1 | https://mathoverflow.net/users/64449 | 250808 | 113,855 |
https://mathoverflow.net/questions/250391 | 1 | For $G$ finite abelian group, let $\eta,\omega:G \times G \to \mathbb{C}^\times$ be group pairings. What can I say about the (non-)degeneneracy of the product pairing $\eta \cdot \omega$ in terms of the (non-)degeneneracy of $\eta$ and $\omega$? The product is defined elementwise: $(\eta \cdot \omega) (g,h):= \eta(g,h)... | https://mathoverflow.net/users/58211 | Non-degeneracy of product of group pairings | This is not an answer, but rather a comment with too many characters to post as a comment.
If $A$ is a finite abelian group and $\chi\colon A\to A^\*$ is an isomorphism, then $\chi$ determines a group isomorphism between $\langle \textrm{End}(A); +,-,0\rangle$ and the abelian group of pairings under pointwise product... | 1 | https://mathoverflow.net/users/75735 | 250813 | 113,857 |
https://mathoverflow.net/questions/250805 | 2 | *A smart man once explained to me how to solve the following problem, then I forgot.*
Let $F\subset\mathbb{R}$
be a number field,
let $d\in F^+$,
and let $K=F(\sqrt{-d})$.
Denote the rings of integers of $F$
and $K$
respectively
by $\mathbb{Z}\_F$
and $\mathbb{Z}\_K$.
Suppose $\mathfrak{p}\vartriangleleft\mathbb{Z}\_... | https://mathoverflow.net/users/14835 | What is the effect of imaginary quadratic extension on a quaternion algebra's ramified primes | The base change of your quaternion algebra will be ramified at $\mathfrak{P}$ if and only the degree of the extension of completions $K\_{\mathfrak{P}}/F\_{\mathfrak{p}}$ has odd degree, i.e. case 3. A general way to see this kind of property is that the invariant in $\mathbb{Q}/\mathbb{Z}$ of the class in the Brauer g... | 9 | https://mathoverflow.net/users/40821 | 250818 | 113,858 |
https://mathoverflow.net/questions/250518 | 2 | Let $\alpha\_0$ be the unique non-trivial character satisfying $\alpha\_0^2=1$ of the split torus $\mathrm{T} \subset \mathrm{SL}(2,q)$ and denote by $\mathrm{R}(\alpha\_0)$ the character of $\mathrm{SL}(2,q)$ obtained by first extending $\alpha\_0$ to the Borel subgroup $\mathrm{B} \subset \mathrm{SL}(2,q)$ and then b... | https://mathoverflow.net/users/98835 | Fields of definition of parabolically induced representations of $\mathrm{SL}(2,q)$ | The answer is yes, the representation is defined over $\mathbb{Q}\_p(\sqrt{q})$. We first claim that $R\_{+}(\alpha\_0)$ and $R\_{-}(\alpha\_0)$ are not isomorphic. This can easily be seen by using the Bruhat decomposition and concluding that $dim(End(R(\alpha\_0)))=2$.
Second, let $\psi$ be the character of $R\_{+}(\... | 2 | https://mathoverflow.net/users/41644 | 250825 | 113,861 |
https://mathoverflow.net/questions/250826 | 31 | My question concerns the proof of the following: Let $a,b,n \in \mathbb{N}$. If $n \neq 1$ and $n$ divides both $a$ and $b$, then $b$ is a composite number or $b$ divides $a$. My proof:
Suppose $b$ is not composite. Then $b$ is prime. Since $n \neq 1$ and $n$ divides $b$, we must have $n = b$. Thus $b$ divides $a$.
... | https://mathoverflow.net/users/40570 | How Would an Intuitionist Prove This? | In general, when working in constructive mathematics, the strategy for proving $Q \lor R$ is to prove $Q$ or to prove $R$. In this case, just knowing abstractly that "there is an $n \not = 1$ that divides both $a$ and $b$" does not directly tell you whether $b$ is composite or whether $b$ divides $a$. So you will need ... | 42 | https://mathoverflow.net/users/5442 | 250830 | 113,863 |
https://mathoverflow.net/questions/250769 | 2 | Suppose $M$ is a complex $n$-dimensioanl compact Kähler manifold and $\omega$ a Kähler class. Suppose $\alpha\in H^{1,1}(M,\mathbb{R})$ is a nef class belonging to the boundary of the Kähler cone of $M$. If for some $1\leq k\leq n-1$ we have
$$\int\_M\alpha^k\omega^{n-k}=0,$$
can we conclude that $\alpha^k=0\in H^... | https://mathoverflow.net/users/36974 | A question about nef classes on compact Kähler manifolds | @Kevin I think the answer is 'yes'. Here is a proof: if $\alpha$ is nef, then, for every $\varepsilon >0$ the class $\alpha+\varepsilon \omega$ is Kahler, and in particular the class $(\alpha+\varepsilon\omega)^k$ contains a positive $(k,k)$-current. We let $\varepsilon$ go to $0$, and obtain in the class $\alpha^k$ a ... | 4 | https://mathoverflow.net/users/48958 | 250847 | 113,870 |
https://mathoverflow.net/questions/250855 | 7 | In some work I've been doing on the cohomology of the moduli space of curves, the following identity has come up:
$$\prod\_{i=1}^n \frac{x^{i-1}}{x^i-1} = \sum\_{(a\_1^{r\_1},\ldots,a\_{\ell}^{r\_{\ell}}) \vdash n} \left(\prod\_{j=1}^{\ell} \frac{1}{a\_i^{r\_i} (x^{a\_i}-1)^{r\_i} (r\_i)!}\right).$$
Here $x$ is a f... | https://mathoverflow.net/users/317 | Identity involving a sum over all partitions of $n$ | Here's a quick sketch (since I'm pressed for time). Multiply both sides of the identity by $t^n$ and sum over $n$ from $0$ to infinity. From the cycle decomposition identity (Polya's formula) the right side becomes
$$
\exp \Big( \sum\_{i=1}^{\infty} \frac{t^i}{i (x^i-1)} \Big)= \exp\Big( -\sum\_{i=1}^{\infty} \frac{t... | 15 | https://mathoverflow.net/users/38624 | 250862 | 113,874 |
https://mathoverflow.net/questions/250854 | 7 | Let $F\_2$ be the free group of rank 2. Let $K\le F\_2$ be a characteristic subgroup, such that $G := F\_2/K$ is finite.
Do there exist examples of such **nonabelian** $G$ such that the induced map
$$Aut(F\_2)\rightarrow Aut(G)$$
is surjective?
For example, if $C\_n$ is the cyclic group of order $n$, then for every... | https://mathoverflow.net/users/88840 | are there finite nonabelian characteristic quotients $G$ of $F_2$ inducing a surjection $Aut(F_2)\twoheadrightarrow Aut(G)$? | I found one example of a verbal subgroup (as suggested by Arturo Magidin) that worked, but others that did not. Let $F/K$ be the largest quotient of $F=F\_2$ that is a $2$-group and has exponent $2$ class $2$. In other words, $K=H^2[F,H]$, where $H=F^2[F,F]$.
Then $|F/K| = 2^5$ and has automorphism group of order $38... | 6 | https://mathoverflow.net/users/35840 | 250867 | 113,876 |
https://mathoverflow.net/questions/250871 | 0 | I have a cubic function:
\begin{equation\*}
h(x)\triangleq \eta+x-\frac{V(\eta-x)^3}{c\eta}
\end{equation\*}
we know that $x\in[0,\eta)$ and all letters are positive and $V>c/\eta$. Hence we know that $h(0)<0$ and $h(\eta)>0$ and $h(x)$ is concave increasing for $x\in[0,\eta)$. So we can infer that there will be exactl... | https://mathoverflow.net/users/140317 | Characterize the Monotonicity of a root of a cubic equation | Differentiating $h(x, \eta) = 0$ implicitly, we get
$$ \dfrac{dx}{d\eta} = \dfrac{-c\eta^2 + V(2\eta+x)(\eta-x)^2}{\eta (3 V (\mu - x)^2 + c \eta)}$$
The denominator is always positive.
On any interval where the numerator doesn't change sign,
$x$ is monotone.
For the numerators of $dx/d\eta$ and $h(x,\eta)$ to ... | 2 | https://mathoverflow.net/users/13650 | 250874 | 113,879 |
https://mathoverflow.net/questions/250880 | 7 | I'm looking for the precise definition of "Lagrangian skeleton", as I'm eventually going to give a talk on this topic. As I asked a professor in my university about references on Lagrangian skeleta, he listed the following:
* Lagrangian Non-intersection by Paul Biran
<http://arxiv.org/pdf/math/0412110v2.pdf>
* From S... | https://mathoverflow.net/users/57191 | Definition of "Lagrangian skeleton" | Indeed the Cieliebak-Eliashberg reference is canonical. I summarize some things here:
The notion of skeleton in symplectic geometry is generally used in the exact setting, i.e. the symplectic form is $\omega = d\lambda$. Note that in this case the symplectic manifold $M$ must be noncompact.
Having fixed the primit... | 12 | https://mathoverflow.net/users/4707 | 250887 | 113,884 |
https://mathoverflow.net/questions/250890 | 21 | The [entry](http://www-history.mcs.st-andrews.ac.uk/Biographies/Vandermonde.html) on Alexandre-Théophile Vandermonde at the MacTutor History
of Mathematics archive ends with the description of the contents of Vandermonde's fourth and last mathematical paper, concluding with the sentence
>
> Finally he gave a rema... | https://mathoverflow.net/users/1409 | Vandermonde's remarkably clever notation for determinants | The history of the Vandermonde notation is described, in the context of the Vandermonde determinant, in section 2.1 of [A case of mathematical eponymy: the Vandermonde determinant](http://arxiv.org/abs/1204.4716) (2010). It seems Lebesgue didn't like it because it could have induced a mix-up between indices and exponen... | 26 | https://mathoverflow.net/users/11260 | 250891 | 113,887 |
https://mathoverflow.net/questions/250817 | 8 | Let $X$ be a Tychonoff space with no isolated points such that the boundary of any subset of $X$ is compact. Does it mean that $X$ is compact ? (If $X$ is a resolvable space then it is clearly compact.)
| https://mathoverflow.net/users/85926 | When the boundary of any subset is compact? | Your property implies in particular:
"Every nowhere dense closed subset of $X$ is compact."
(A nowhere dense closed subset is the boundary of its complement.)
This in turn is equivalent to compactness for $T\_1$ spaces with no isolated points, as shown by Katetov in 1947. Interestingly, the result holds as well f... | 10 | https://mathoverflow.net/users/29491 | 250893 | 113,888 |
https://mathoverflow.net/questions/250869 | 6 | Let $E,F$ be complex vector bundles over some closed manifold $M$. We investigate operators $T:C^{\infty}(M,E) \to C^{\infty}(M,F)$ between smooth sections of these bundles. We say that such operator is order zero differential operator if $[T,f]=0$ this means that $T$ is a bundle endomorphism. If we know what is order ... | https://mathoverflow.net/users/98866 | Abstract definition of differential operators | First of all, you can show that operators defined by your recursive rule are local operators, i.e., if a section $s$ has support in an open set $U$ then $T(s)$ has support in $U.$ By using the partition of unity it remains to do the work locally. As already mentioned by Simon Henry in his comment, when the order $k=1$,... | 5 | https://mathoverflow.net/users/4572 | 250899 | 113,890 |
https://mathoverflow.net/questions/250886 | 1 | Let $G$ be a semisimple group over $\mathbb{C}$ and let $X=G/H$ be a spherical homogeneous space,
then $X$ defines a spherical datum (Luna datum) $\mathcal L(X)=(N,\mathcal V, \mathcal D, \rho,\varsigma)$, see below.
The spherical datum is a functor. Indeed, assume we have two spherical homogeneous spaces $X\_1$ and ... | https://mathoverflow.net/users/4149 | Morphisms of the spherical data of spherical homogeneous spaces | The situation is more complicated. First of all $\lambda\_{\mathcal D}$ does not exist, since there may be colors in $X\_1$ which map dominantly to $X\_2$. The best way to describe morphisms $X\_1\to X\_2$ is to use some kind of Stein factorization $X\_1\to X'\to X\_2$ where $X\_1\to X'$ has connected fibers and $X'\to... | 3 | https://mathoverflow.net/users/89948 | 250902 | 113,891 |
https://mathoverflow.net/questions/250912 | 9 | What is an example of a functor $$F : \mathcal{C} \to \mathcal{D}$$ between two Grothendieck toposes which preserves colimits and finite products, but is not left exact (i.e., does not preserve pullbacks)?
I just assume that there is such an example, since otherwise the notion of an algebraic morphism between toposes... | https://mathoverflow.net/users/98306 | Cocontinuous product-preserving functor between Grothendieck toposes | For any small category $J$, the colimit functor $\mathsf{Set}^J \to \mathsf{Set}$ preserves colimits. It preserves finite limits if and only if $J$ is filtered and it preserves finite products if and only if $J$ is sifted. So we only need an example of a sifted but non-filtered category. One such example is $\Delta^{\m... | 16 | https://mathoverflow.net/users/12547 | 250914 | 113,894 |
https://mathoverflow.net/questions/249533 | 2 | Apart from the Deligne's original paper "Le déterminant de la cohomologie", is there any other reference presenting the construction of the Deligne pairing on arithmetic surfaces?
Of course there is "nothing wrong" with the original paper, I'm only looking for other sources.
| https://mathoverflow.net/users/47136 | Alternative reference to Deligne pairing | See Arbarello/Cornalba/Griffiths: "Geometry of algebraic curves II" p. 366-379.
| 4 | https://mathoverflow.net/users/61532 | 250916 | 113,895 |
https://mathoverflow.net/questions/250908 | 5 | Given two (Euclidean or hyperbolic) triangles $ T = ABC $ and $ T' = A'B'C' $,
the natural map is the one that sends
$ A' \mapsto A $,
$ B' \mapsto B $,
$ C' \mapsto C $
and maps affinely each side of $T'$
onto the corresponding side of $T$.
We say that the triangle $ T' $ dominates
the triangle $ T $
if the nat... | https://mathoverflow.net/users/91134 | Short map between hyperbolic triangles | No, this is not true. Here is an indirect argument (if I made no mistake).
If the statement would be true for $\epsilon > 0$ small enough, then it would be true for all $\epsilon > 0$ (since the maps commute, the set of "good" epsilons for a given $(a,b,c)$ is open; this set is also closed because the Lipschitz condi... | 5 | https://mathoverflow.net/users/98590 | 250927 | 113,899 |
https://mathoverflow.net/questions/250925 | 1 | I'm trying to understand why every Pfaffian cubic fourfold contains a rational normal quartic scroll.
I believe this is a well-known classical construction (for example, see Hassett's paper on "special cubic fourfolds"), but I haven't been able to find a reference explaining explicitly how to construct the quartic sc... | https://mathoverflow.net/users/nan | Why do Pfaffian cubic fourfolds contain quartic scrolls? | The construction I know is somewhat indirect. Let $X\subset \mathbb{P}^5$ be defined by the pfaffian of a skew-symmetic matrix $A$ of linear forms. This gives an exact sequence
$$0\rightarrow \mathcal{O}\_{\mathbb{P}}(-1)^6\xrightarrow{\ \ A\ \ } \mathcal{O}\_{\mathbb{P}}^6\rightarrow E\rightarrow 0$$ where $E$ is a ra... | 1 | https://mathoverflow.net/users/40297 | 250931 | 113,900 |
https://mathoverflow.net/questions/250858 | 8 | Let $\sqsubset=\bigcup\_n \sqsubset\_n$ be a relation on $\omega^\omega$ where each $\sqsubset\_n$ is arithmetic and $\{f: f\sqsubset\_n g\}$ is closed for each $g\in \omega^\omega, n\in \omega, i.e. \Pi\_1^0(g)$, a typical example is domination past $n$.
In the following fix $\chi $ large enough regular cardinal.
... | https://mathoverflow.net/users/23835 | Counter-example that "almost preserving" is not preserved under countable support iteration | We can do something like setting $f\sqsubset\_n g$ if and only if $f(m)=g(n^m)$ for all $m$ (so $f$ looks like $g$ restricted to the powers of $n$).
For each $g$, the set $\{f:f\sqsubset\_n g\}$ is a singleton, hence closed.
Also, given countably many $f$ we can find a single $g$ covering them all using powers of pri... | 7 | https://mathoverflow.net/users/18128 | 250935 | 113,902 |
https://mathoverflow.net/questions/250928 | 7 | Let $\mathcal{M}\_{g,1}\to \mathcal{M}\_g$ be the universal genus $g$ curve, let $K\_g$ denote the funciton field of $\mathcal{M}\_g$. Take the generic fiber $C\to \mathrm{Spec}{(K\_g)}$, after some finite extension $L/K\_g$, $C\_L$ will admit a rational point. Is there a good choice for $L$? If there is, do we know $\... | https://mathoverflow.net/users/nan | Section of universal curve | Let me assume $g>2$, so that there is a curve $C$ over $K\_g$ of genus $g$ (the *generic curve*) -- the genus condition assures that the generic curve has no automorphisms. It is known but not obvious that $C$ has no $K\_g$-rational points (this is due to Hain-Matsumoto in characteristic zero and Watanabe in positive c... | 6 | https://mathoverflow.net/users/6950 | 250939 | 113,906 |
https://mathoverflow.net/questions/250951 | 2 | Suppose we have a family $F$ such that:
1. For each $A \in F$ we have $|A| = k$
2. For each $A,B \in F$ we have $A \cap B \neq \emptyset$
3. If we have $C$ such that for each $A \in F$ we get $C \cap A \neq \emptyset$ then $|C| \ge k$
***Question***: For a fixed $k$, can $F$ be arbitrarily large? Can $F$ be infinit... | https://mathoverflow.net/users/59012 | Finite limit to the size of an intersecting family of k-sets with no smaller intersecting set? | It is finite. Assume that we managed to find $k+1$ sets so that intersection of any two of them is the same set $C$. Then this $C$ satisfies 3, a contradiction. If we have many sets, such $k+1$ (or as many as you wish) sets may be always found, this is a [Sunflower](https://en.wikipedia.org/wiki/Sunflower_(mathematics)... | 3 | https://mathoverflow.net/users/4312 | 250952 | 113,908 |
https://mathoverflow.net/questions/250948 | 3 | Let $A$ and $B$ be nonempty finite sets of cardinalities $n$ and $m$, respectively. The *distance* between two functions $f, g : A \to B$ is defined as the number of disagreements between them, that is,
$$
d(f,g) := \# \{x\in A : f(x) \neq g(x)\}.
$$
Let $ 0 < k < n$. A set $F \subset B^A$ is called *$k$-separated* ... | https://mathoverflow.net/users/1516 | Separated sets of functions between finite sets | Expanding on the comments of Andreas and Fedor I will point out a resource where this data is kept track of. This will give some measure of how unknown these numbers are (and when they are actually known).
For $m=2$ you are looking for the largest binary code of length $n$ with minimum distance $k$. Which as Fedor sa... | 2 | https://mathoverflow.net/users/51668 | 250963 | 113,911 |
https://mathoverflow.net/questions/250946 | 18 | Let $J\_k$ be a $k \times k$ all ones matrix and $B$ **any** $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$.
I would like to show that the matrix $$X\_B = (J\_k -I) - B (J\_k - I)^{-1} B^T\,,$$ is not positive-definite. In other words, I'd like to show that
>
> At least one eigenvalue o... | https://mathoverflow.net/users/1737 | Showing that a certain matrix is not positive definite | Counterexample: let $k=7$, and let $B$ be the circulant matrix with $B\_{ij}=1$ **iff** $i-j \in \{1,2,4\} \bmod 7$. Then $X\_B$ is $I + \frac12 J$, with characteristic polynomial $(x-1)^6 (x-\frac92)$. Or use $B+I$ instead to get $I + \frac13 J$, with characteristic polynomial $(x-1)^6 (x-\frac{10}{3})$.
| 17 | https://mathoverflow.net/users/14830 | 250966 | 113,912 |
https://mathoverflow.net/questions/250965 | 5 | Let $X$ be a smooth compact oriented 4-manifold with $\partial X=L(p,1)$, $H\_2(X;\Bbb Z)=\Bbb Z$, $H\_3(X; \Bbb Z)=0$ and the induced map $\pi\_1(L(p,1)) \to X$ surjective. What are the possibilities for $\pi\_1(X)$? In particular, are there examples where $\pi\_1 \ne 0$?
| https://mathoverflow.net/users/50754 | $\pi_1$ of 4-manifolds that "look like" disk bundles | Let's take your $p$ to be prime. Then $X$ has to be simply connected, even without all of the hypotheses. Here is the argument.
From the map on $\pi\_1(L) \to Z\_p$ you get a map $L \to BZ\_p$. This map is clearly $0$ in $H\_3$, since it factors through the inclusion of $L$ into the $4$-manifold $X$. On the other ha... | 3 | https://mathoverflow.net/users/3460 | 250974 | 113,914 |
https://mathoverflow.net/questions/250972 | 3 | We use the following definition of almost periodicity given in <https://arxiv.org/pdf/math-ph/0005018.pdf>
Given a bounded function $f :\mathbb Z \to\mathbb R$ we denote the set of translates of $f$ by $U\_0$. The function $f$ is said to be almost periodic if $U\_0$ is precompact in $\ell\_\infty(\mathbb Z)$.
For i... | https://mathoverflow.net/users/20838 | Is $f(n)=\cos(2\pi\theta(1+2+\ldots+n))$ almost periodic? | I believe the answer is **no**, for the same sort of reason that $g(t)=e^{it^2}$ is not almost periodic on $\mathbf R$.
(Indeed, the latter would mean that given $\varepsilon>0$ there is a $T$ such that every interval of length $T$ contains an "$\varepsilon$-almost period" of $g$, that is, a number $s$ such that $\su... | 3 | https://mathoverflow.net/users/19276 | 250977 | 113,915 |
https://mathoverflow.net/questions/250836 | 2 | As is well known, using the Hilbert Nullstellensatz (and a more recent result of Cartier) one can show that commutative finitely generated Hopf algebras over $\mathbb{C}$ are equivalent to algebraic groups (cf the accepted [answer](https://mathoverflow.net/questions/9046/hopf-algebras-arising-as-group-algebras) of Davi... | https://mathoverflow.net/users/95346 | Finitely Generated Commutative Hopf $*$-Algebras | I think you have to assume that your commutative Hopf $\*$-algebra is spanned by the matrix coefficients of *unitary* corepresentations (in which case that's the case: [1] or [2, Theorem 1.6.7] to get Woronowicz compact quantum group and one can use the Gelfand-Naimark duality). For example, take $H$ to be the group al... | 2 | https://mathoverflow.net/users/9942 | 250987 | 113,917 |
https://mathoverflow.net/questions/250995 | 4 | The question is the title. The set $Ba(C(K))$ is the unit ball of $C(K)$. This has to be known, but I can't find the answer explicitly in the literature. There is some literature about polyhedral Banach spaces however I'm not sure if being polyhedral is sufficient to show the set of extreme points of the ball is counta... | https://mathoverflow.net/users/15388 | If $K$ is a countable compact metric space is the set of extreme point of $Ba(C(K))$ countable? | It is true (if you mean real spaces, for complex spaces it is obviously false). Note that if a function $f$ is extremal, it takes only values $\pm 1$. Indeed, if $|f(a)|<1$, then choose a small ball $B(a,r)$ on which $|f|<1-r$ and consider the functions $f(x)\pm \max(r-d(a,x),0)$. They belong to a unit ball and are dif... | 8 | https://mathoverflow.net/users/4312 | 250998 | 113,918 |
https://mathoverflow.net/questions/250954 | 0 | Let $G$ be a finite simple group, and $F$ a profinite group (I'm really interested in the case where $F$ is free of finite rank, in particular rank 2).
In Ribes-Zalesskii, they define the $G$-rank of $F$ to be the integer $n\_G$ such that if $K\_G$ is the intersection of all open normal subgroups of $F$ whose quotien... | https://mathoverflow.net/users/88840 | Difference between $G$-rank and the maximal $G$-power quotient | Your first thought was correct: the number $n\_G$ is $m\_G$.
More precisely, if $F$ is any finitely generated (profinite) group and $G$ is a finite simple group. Let $T\_G(F)$ be the set of all normal subgroups $N$ in $F$ such that $F/N$ is isomorphic to $G$. Write $K\_G = \bigcap\_{N \in T\_G(F)} N$. As you mentione... | 2 | https://mathoverflow.net/users/54441 | 251009 | 113,921 |
https://mathoverflow.net/questions/250985 | 4 | Let $X$ be a finite CW-complex such that its $K$-theory $K^\*(X)$ is, as a $\mathbb{Z}$-algebra, generated by $a\_1, \cdots, a\_n$ which are represented by reduced line bundles $L\_1-1, \cdots, L\_n-1$ satisfying $L\_i^{\otimes 2}\oplus 1\cong L\_i^{\oplus 2}$ (implying that $a\_i^2=0$) for $1\leq i\leq n$. Note that $... | https://mathoverflow.net/users/85722 | Atiyah-Hirzebruch spectral sequence for a special kind of CW-complexes | This is not true. One of the main reasons is that, because you've only specified properties about $K^\*(X)$, that leaves a lot of room for $H^\*(X;\Bbb Z)$ to have information which doesn't ultimately contribute to $K$-theory.
Finding an example is a little more work. The mod-2 Moore space $M(\Bbb Z/2, n)$ is the $(n... | 8 | https://mathoverflow.net/users/360 | 251011 | 113,922 |
https://mathoverflow.net/questions/250997 | 1 | It is well-known that the curvature forms of the (complexified) Levi-Civita connection can be used to provide explicit representatives for Chern classes of compact Kähler manifolds. This is not true for symplectic manifolds. So my question is, for a general compact symplectic manifold, does there exist any method to pr... | https://mathoverflow.net/users/36974 | Representatives of Chern classes for compact symplectic manifolds | The symplectic manifolds equipped with an adapted almost complex structure are also called almost-Kahler manifolds. There are many connections on the tangent bundle compatible with the metric induced by the symplectic form and the almost complex structure. Each one of them is uniquely determined by its torsion.
Accor... | 4 | https://mathoverflow.net/users/20302 | 251020 | 113,924 |
https://mathoverflow.net/questions/251012 | 6 | Let $M$ be a differentiable manifold of dimension $n>2$ with a Riemannian metric $g=\sum\_{i,j=1}^ng\_{ij}dx\_idx\_j$ such that in some points on $M$ its coefficients $g\_{ij}$ are not differentiable (so $g\_{ij}$ are just continuous on $M$). Call $d\_g$ the metric induced on $M$ by $g$ (by the infimum of the lengths o... | https://mathoverflow.net/users/nan | Geodesics for non differentiable riemannian metric | This is a length-metric, that is any two points $x$ and $y$ can be joined by a path with length arbitrary close to the distance from $x$ to $y$.
Further, your metric space is locally bi-Lipschitz to the Euclidean space,
in particular it is locally compact.
If your space is complete then by [Hopf–Rinow theorem](http:/... | 7 | https://mathoverflow.net/users/1441 | 251021 | 113,925 |
https://mathoverflow.net/questions/251018 | 7 | What is the low degree cohomology of the mapping class group of a non-orientable surface? More specifically, what is the universal central extension of the mapping class group of a non-orientable surface?
(I've done some googling, but so far have only found presentations of the MCG.)
(I would be happy to have the a... | https://mathoverflow.net/users/284 | Cohomology of the mapping class group of a non-orientable surface? | Let me write $\mathcal{N}\_g$ for the mapping class group of the connect sum of $g$ projective planes. Nathalie Wahl proved that these groups enjoy homological stability, and in
>
> O. Randal-Williams, The homology of the stable non-orientable mapping
> class group, Algebraic & Geometric Topology 8 (2008) 1811-18... | 12 | https://mathoverflow.net/users/318 | 251023 | 113,926 |
https://mathoverflow.net/questions/217719 | 22 | Function $y(x)$ on the plane defines a conic (strictly speaking, at most second order algebraic curve) if and only if $\frac{d^3}{dx^3} (y'')^{-2/3}=0$. What is intuition behind this? How may I see without calculations that this equation is symmetric in $x$, $y$, or, say, that it is affine invariant? Quick calculation ... | https://mathoverflow.net/users/4312 | differential equation of conics | I didn't see this question when it originally appeared, but an edit today brought it to the front page of "new questions" where I saw it. Back around 2005 or 2006 I came across this differential equation for conics (see this [26 April 2008 sci.math post](https://groups.google.com/g/sci.math/c/LwkcylTv_F8/m/66A8lWa1UfcJ... | 15 | https://mathoverflow.net/users/15780 | 251036 | 113,930 |
https://mathoverflow.net/questions/251034 | 1 | **Motivation**
Then the usual stochastic filtering problem says that:
$$
\operatorname{argmin}\_{Z \in L^2(\mathscr{G}\_t)}\,\mathbb{E}[(Y\_t-Z\_t)^2],
$$
where $\mathscr{G}\_t$ is the $\sigma$-algebra generated by $Y$ up to time $t$ is solved by
$$
\hat{Y}\_t\triangleq \mathbb{E}[Z|\mathscr{G}\_t].
$$
---
**Qu... | https://mathoverflow.net/users/36886 | MSE of measurable function is still conditional expectation | The time dependence is just muddying the waters. The question is really, given a $\sigma$-field $\mathcal{G}$, is $\hat{Y} = \mathbb{E}[Z \mid \mathcal{G}]$ the minimizer of $\mathbb{E}[(\phi(Y) - \phi(Z))^2]$ over $\mathcal{G}$-measurable random variables?
Clearly not. Work in $d=d'=1$, let $\mathcal{G}=\{\Omega, \e... | 3 | https://mathoverflow.net/users/4832 | 251040 | 113,931 |
https://mathoverflow.net/questions/251042 | 2 | Most sources about motivic homotopy theory mention that the category of (smooth) separated schemes of finite type over a (Noetherian of finite Krull dimension) base $S$ is essentially small, which is essential for the homotopy theory of its simplicial sheaves. Yet, none of the sources I came across proves or references... | https://mathoverflow.net/users/24453 | Smallness of the category of schemes of finite type | You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows:
* If $S$ is affine and we only consider affine finite type $S$-schemes, these correspond to finite type $\Gamma(S)$-algebras. These are isomorphic to algebras of the form $\... | 12 | https://mathoverflow.net/users/98306 | 251044 | 113,933 |
https://mathoverflow.net/questions/251010 | 2 | Consider the space $\mathrm{SU}(2)^\natural$ of conjugacy classes in $\mathrm{SU}(2)$. It has a natural identification with the interval $[0,\pi]$ with Haar measure $\frac{2}{\pi} \sin^2\theta\, \mathrm{d}\theta$, via the mapping
$$
\theta \mapsto x\_\theta = \begin{pmatrix} e^{i\theta} \\ & e^{-i\theta}\end{pmatrix} ... | https://mathoverflow.net/users/6856 | Variation of trace of symmetric powers | For the large-$k$ asymptotics I would first approximate $|U'\_k(\theta)|$ by its envelope
$$F\_k(\theta)=\frac{k}{\sqrt{2}\sin\theta},$$
plotted together for $k=50$:
The divergence of $F\_k(\theta)$ at $\theta=0$ is cut-off at $\theta\_1=1/k$, and similarly the di... | 4 | https://mathoverflow.net/users/11260 | 251045 | 113,934 |
https://mathoverflow.net/questions/249868 | 19 | The sum of two nilpotent elements of a commutative ring is nilpotent. This can be checked by a direct calculation using the binomial theorem. In fact, this calculation shows the stronger statement $x^n=y^m=0 \Rightarrow (x+y)^{n+m-1}=0$.
But we can also give a more sophisticated proof: If $x,y$ are nilpotent, they ar... | https://mathoverflow.net/users/98306 | How to construct a constructive proof from a non-constructive proof using prime ideals? | Since this is somewhat hidden in the comments, let me give the following answer:
* The statement that the sum of two nilpotents is nilpotent is so basic that it seems to be used in the construction and the verification of the Zariski locale/topos/lattice. I don't think that constructive algebra can prove this without... | 3 | https://mathoverflow.net/users/98306 | 251051 | 113,935 |
https://mathoverflow.net/questions/251064 | 2 | I am reading the book *"Stochastic Optimal Control: The Discrete Time Case", by Bertsekas and Shreve* (hereafter called "**the Book**"), and I recently observed that a statement made in page 10 of the book (Introduction) seems that can be stated somewhat more generally.
The statement under question is described in th... | https://mathoverflow.net/users/99108 | Existence of ε-optimal Borel measurable policies in stochastic control | Your claim isn't true.
It's known that there exists a Borel subset $A \subset \mathbb{R} \times \mathbb{R}^+$ whose projection onto the first coordinate $B = \pi\_1(A) = \{ x : \exists y ((x,y) \in A)\}$ is not Borel. Let $$g(x,y) = \begin{cases} 0, & y > 0, (x,y) \in A \\ 1, & y > 0, (x,y) \notin A \\ \frac{1}{2}, &... | 2 | https://mathoverflow.net/users/4832 | 251066 | 113,939 |
https://mathoverflow.net/questions/250728 | 10 | Fano varieties are defined by the ampleness of $-K\_X$, and a rough statement of a step in the Mori program is to check whether a variety is a Fano-fibered one. By that reason, Fano ones are important in classification problems.
From somewhere, I have read the phrase "Fano varieties have rich geometry". What is geome... | https://mathoverflow.net/users/75699 | Why do we say fano varieties have rich geometry? | Let $X$ be an smooth Fano variety of dimension $n$, defined over $\mathbb{C}$ to simplify.
Here is a non exhaustive list of some nice properties of Fano varieties, which of course can be complemented by the experts here in mathoverflow.
1. From Kodaira vanishing theorem, $\operatorname{H}^i(X,\mathcal{O}\_X)=0$ fo... | 8 | https://mathoverflow.net/users/31724 | 251075 | 113,942 |
https://mathoverflow.net/questions/251058 | 5 | The well known *Ramsey number* $R(k)$ is the least integer $n$ so that every 2-edge coloring of $K\_n$ contains a monochromatic $K\_k.$
Another interpretation of the above definition is that every graph on $R(k)$ vertices has a $K\_k$ or $\overline{K\_k}$ as a (induced) subgraph. There are many generalizations of Ram... | https://mathoverflow.net/users/1737 | A variant of Ramsey numbers | There is a conjecture of Erdős, Fajtlowicz and Staton which is closely related to your question. Define $\hat{R}(k)$ to be the smallest positive integer $n$ such that any graph on $n$ vertices contains an induced regular subgraph on $k$ vertices. Note that this class includes complete graphs, independent sets and compl... | 7 | https://mathoverflow.net/users/66275 | 251082 | 113,943 |
https://mathoverflow.net/questions/251063 | 3 | I have distance matrix $D$ that was calculated by some distance (non-Euclidean but satisfying distance requirements). Is there a set of points in some Euclidean space such that it generates matrix of Euclidean distances that is equal $D$?
I know that if $G=-HDH/2$ is p.d. where H is the centering matrix then such emb... | https://mathoverflow.net/users/99111 | Given a distance matrix is there an isometric embedding? | No. The magic words are [Cayley-Menger](http://mathworld.wolfram.com/Cayley-MengerDeterminant.html), but an explicit answer is given [here.](https://mathoverflow.net/questions/12394/representability-of-finite-metric-spaces)
| 4 | https://mathoverflow.net/users/11142 | 251089 | 113,946 |
https://mathoverflow.net/questions/251070 | 5 | Let $d$ be a positive integer and $0 \leq k \leq 3d-2$ another integer.
**Question 1.** If $p\_1,...,p\_k$ are $k$ *general* points in ${\mathbb P}^2$, is it known in general whether the space parametrizing degree $d$ plane rational curves passing through $p\_1,...,p\_k$ is irreducible?
I think it's clear that the ... | https://mathoverflow.net/users/99113 | Plane curves through a given collection of points | Yes, for every integer $k$ with $0\leq k \leq 3d-2$, the following evaluation morphism is surjective and the generic fiber is geometrically irreducible, $$\text{ev}\_{1,2,\dots,k}:\overline{\mathcal{M}}\_{0,k}(\mathbb{P}^2,d)\to (\mathbb{P}^2)^k.$$
One proof uses the version of Bertini's connectedness theorem as formul... | 3 | https://mathoverflow.net/users/13265 | 251090 | 113,947 |
https://mathoverflow.net/questions/251002 | 2 | Let $B$ be an indefinite quaternion algebra over the rationals, let $G$ be the reductive algebraic group defined by $G(A) = (B\otimes A)^\*$ for ${\bf Q}$-algebras $A$; hence $G({\bf R}) = GL\_2({\bf R})$ in particular. Then $(G, {\bf H}^{\pm})$ is a Shimura datum (${\bf H}^{\pm}$ is a the union of lower and
upper hal... | https://mathoverflow.net/users/42721 | Centralizer of Shimura datum defining a Shimura curve in $A_2$ | There is a piece of data missing from the question. As explained in [this question](https://mathoverflow.net/questions/45704/shimura-datum-of-family-of-fake-elliptic-curves?rq=1), in order to get the embedding of Shimura data, you have to choose an element $\gamma \in B^\times$ such that $\gamma^\* = -\gamma$ and $\gam... | 3 | https://mathoverflow.net/users/2481 | 251092 | 113,948 |
https://mathoverflow.net/questions/250970 | 2 | In Alex Lee's undergraduate thesis (2000), it was said that the Hilbert scheme $H\_{2m+2}(\mathbb{P}^3)$ has two components $\mathcal{H}',\mathcal{H}''$, where a general point of $\mathcal{H}'$ corresponds to a pair of skew lines and a general point of $\mathcal{H}''$ corresponds to a conic union a point. At the time o... | https://mathoverflow.net/users/16356 | Hilbert scheme of a plane conic union a point | I think [this paper](https://arxiv.org/pdf/0911.2221v3.pdf) by Chen and Nollet is relevant. Theorem 1.9 (proved as Theorem 4.3 on p. 16) shows the following about plane curves in $\mathbb P^3$.
>
> The component $H\_d\subset \textrm{Hilb}^{dz+2-g}(\mathbb P^3)$ whose
> general point is a degree $d$ plane curve uni... | 1 | https://mathoverflow.net/users/97902 | 251094 | 113,949 |
https://mathoverflow.net/questions/251103 | 7 | I asked [here](https://math.stackexchange.com/questions/1947445/principal-ideal-ring-does-there-exist-a-unit-matrix-such-that-certain-matrix-is) on Math Stack Exchange the following question.
>
> Let $R$ be a principal ideal ring. If $A$ is any $p \times q$ matrix over $R$, then does there exist an invertible matri... | https://mathoverflow.net/users/99127 | Principal ideal ring, does there exist an invertible matrix such that certain matrix is upper triangular? | The answer to your initial question is yes, and Igor Rivin's comment is right and good. The result follows from the fact that a commutative *PIR* (Principal Ideal Ring) with identity is a Hermite ring in the sense of I. Kaplansky.
Indeed, a Hermite ring possesses a trigonal reduction as required, cf. [1, Theorem 3.5]... | 9 | https://mathoverflow.net/users/84349 | 251107 | 113,956 |
https://mathoverflow.net/questions/251110 | 5 | Let $R$ be a commutative ring spectrum (interpret this as you will; as an $E\_\infty$-ring or as a commutative $S$-algebra etc.) and $\operatorname{GL}\_1(R)$ as usual denote its space of units. If $\tilde R$ is the connective cover of $R,$ is there a simple relationship between $\operatorname{GL}\_1(\tilde R)$ and $\o... | https://mathoverflow.net/users/39713 | Group of units of a ring spectrum vs of its connective cover | By definition, the space $GL\_1(R)$ is the subspace of $\Omega^\infty R$ consisting of those elements whose path component $\alpha \in \pi\_0(\Omega^\infty R) = \pi\_0(R)$ is a unit in $\pi\_0(R)$.
If $\tilde R \to R$ is a connective cover, then the map of spaces $\Omega^\infty \tilde R \to \Omega^\infty R$ is an iso... | 12 | https://mathoverflow.net/users/360 | 251111 | 113,957 |
https://mathoverflow.net/questions/251091 | 2 | Let $C \in \mathfrak{gl}(\mathbb{Z},n)$ be a symmetric full rank integer valued matrix (in my case its the symmetric part of a Cartan matrix).. Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice, s.t. $C\mathbb{Z}^n \subseteq\Lambda \subseteq \mathbb{Z}^n$. One can think of $\Lambda$ as a lattice in between... | https://mathoverflow.net/users/58211 | Intermediate lattices $C\mathbb{Z}^n \subseteq \Lambda \subseteq \mathbb{Z}^n$ | First of all, every full rank sublattice $\Lambda\subseteq \mathbb Z^n$ contains $\Lambda'=c\mathbb Z^n$ for some positive integer $c$. This $\Lambda'$ may serve as $C\mathbb Z^n$; so the question can in fact be reformulated as follows: *Does **every** full rank sublattice $\Lambda\subseteq \mathbb Z^n$ have a symmetri... | 5 | https://mathoverflow.net/users/17581 | 251114 | 113,959 |
https://mathoverflow.net/questions/251109 | 10 | Would it be possible to use a paraconsistent logic and axioms similar to ZFC to create a formal sytem, that can be proven to be non-trivial (so that there are some statements which can´t be proven in the system), and which can serve as a foundation for mathematics?
Because it´s possible that the current ZFC + first o... | https://mathoverflow.net/users/99129 | Is a paraconsistent and provably non-trivial foundation for math possible? | There's basically no reason to think that ZFC is inconsistent. Even if it is inconsistent, there's an attractive fragment called [Peano arithmetic](https://en.wikipedia.org/wiki/Peano_axioms), which seems self-evidently consistent to me. Gödel's incompleteness still holds here, and it holds in even weaker fragments of ... | 5 | https://mathoverflow.net/users/3711 | 251116 | 113,960 |
https://mathoverflow.net/questions/251027 | 3 | The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:
Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[0,T]$. We have a given set of admissible controls $\mathcal A$ which take values in a set $U\subset\mathbb R$ and we c... | https://mathoverflow.net/users/85330 | Generalizing HJB equation for a terminal stopping time | At least formally, the given HJB equation with possibly some additional Dirichlet boundary conditions (more on this point below) does hold for the value function associated to the stopped or absorbed process. Recall that the main tool in the (formal) derivation of HJB is *Ito's change of variables formula for semimarti... | 2 | https://mathoverflow.net/users/64449 | 251122 | 113,964 |
https://mathoverflow.net/questions/184356 | 4 | I am trying to identify or find the ordinary or rational generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted $$b(1;n,k)=b(n,k)$$
These numbers are the number of ways to partition a set of $n$ elements into $k$ disjoint parts whose partition cardi... | https://mathoverflow.net/users/60457 | Ordinary or Rational Generating Function for Associated Stirling Numbers $b(n,k)$ | **Yes** , the series $\beta\_k(x):=\sum\_{n\ge0} b(n,k) x^n$ are indeed **rational functions**, with the poles as you said.
*One step back.* The Exponential Generating Function of the polynomials $\big\{\sum\_{k\ge0} b(n,k)t^k \big\}\_{n\ge0}$ is
$$\sum\_{n\ge0}\Big(\sum\_{k\ge0} b(n,k) t^k \Big)\frac{x^n}{n!} =e^{t... | 2 | https://mathoverflow.net/users/6101 | 251123 | 113,965 |
https://mathoverflow.net/questions/250236 | 2 | Prove that for given $a>0$ there exists $C(a)$ depending on $a$ such that $$\sum\_{n\leq x}\Big(\frac{n}{\phi{(n)}}\Big)^a\leq C(a)x$$where $\phi(n)$ denotes Euler totient function.
| https://mathoverflow.net/users/97687 | An estimate on the summatory function of $\Big( \frac{n}{\phi(n)} \Big)^a$ | 1. Show that $$\left( \frac{n}{\phi(n)} \right)^a =\prod\_{p \mid n} \left(1 + \frac{1}{p-1} \right)^a \le \prod\_{p \mid n} \left(1 + \frac{C}{p} \right) = \sum\_{d \mid n} \frac{\mu(d)^2}{d} C^{\Omega(d)}$$
for some explicitly computable constant $C$ depending on $a$.
2. Bound your sum from above by
$$\sum\_{d \le x}... | 3 | https://mathoverflow.net/users/31469 | 251127 | 113,967 |
https://mathoverflow.net/questions/251152 | 4 | On pp.78 of these [notes](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_Knotes.pdf) live TEX-ed by Arun Debray for Dan Freed's K-theory [course](https://www.ma.utexas.edu/users/dafr/M392C/) (lecture 23 given by Andrew Blumberg), there is a comment about how Hyman Bass initially started from the topologic... | https://mathoverflow.net/users/82645 | Analogy behind Hyman Bass' definition of algebraic $K_1$ | For a space $X$, isomorphism classes of rank-$k$ vector bundles on the suspension $SX$ are in one-one correspondence with homotopy classes of maps from $X$ to $GL\_k(F)$, where $F$ is ${\mathbb R}$ or ${\mathbb C}$. You get $K\_1(X)$ by replacing $GL\_k(F)$ with the direct limit $GL(F)$. The algebraic analogue, then, i... | 8 | https://mathoverflow.net/users/10503 | 251156 | 113,977 |
https://mathoverflow.net/questions/250859 | 6 | In my recent work I've become interested in working with the minimizer of
$$
\mathbb{E}[(Y-Z)^2] + \lambda P(Z),
$$
$Y$ is an observed random variable, $P$ is a positive-convex penalty function, $Z$ is a measurable random variable with respect to the $\sigma$-algebra generated by $Y$ and $\lambda\geq 0$.
If $\lambda... | https://mathoverflow.net/users/36886 | Does there exist a Penalized Conditional Expectation? | This is a bit different and doesn't address the question, but hopefully close enough to be useful: we know some things about $\mathbb{E} L(Z,Y)$ for other loss functions $L$.
If and only if $L$ is a Bregman divergence, the mean is the minimizer. This is attributed to Banerjee et al. 2005; try [Mark Reid's blog post](... | 3 | https://mathoverflow.net/users/29697 | 251162 | 113,978 |
https://mathoverflow.net/questions/250814 | 22 | For which integers $n$ does every surjection $SL\_2(\mathbb{Z})\twoheadrightarrow SL\_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$?
(this is the usual kernel, ie, the subgroup of matrices congruent to 1 mod $n$).
A positive answer (for some $n$) would certainly imply that $\Gamma(n)$ is characteristic inside ... | https://mathoverflow.net/users/88840 | For which $n$ is it true that all surjections $SL_2(\mathbb{Z})\rightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ have kernel $\Gamma(n)$? | The following mostly answers the question:
**Claim 1**: If every surjection $SL\_{2}(\mathbb{Z}) \to SL\_{2}(\mathbb{Z}/n\mathbb{Z})$ has kernel $\Gamma(n)$, then all prime factors $p | n$ have $p \leq 11$.
To prove Claim 1, I need another claim.
**Claim 2**: If $p > 11$ is prime, there are elements $x, y \in PSL... | 9 | https://mathoverflow.net/users/48142 | 251165 | 113,980 |
https://mathoverflow.net/questions/251033 | 3 | The following concept seems to be useful:
>
> **Definition.** Let $\mathbf{J}$ and $\mathbf{C}$ denote categories, and suppose we're given a functor $F:\mathbf{J} \rightarrow \mathrm{End}(\mathbf{C}).$ A *generalized cocone* from $F$ is an object $X$ of $\mathbf{C}$ together with, for each object $Y$ of $\mathbf{J}... | https://mathoverflow.net/users/26080 | What are generalized cocones/colimits really called? | I don't know references. But here is a reformulation of this notion which might be useful.
I will assume that $\mathbf{C}$ is small-cocomplete, $\mathbf{J}$ is small and that $F : \mathbf{J} \times \mathbf{C} \to \mathbf{C}$ is a functor (this is equivalent to a functor $\mathbf{J} \to \mathrm{End}(\mathbf{C})$). The... | 5 | https://mathoverflow.net/users/98306 | 251171 | 113,985 |
https://mathoverflow.net/questions/251159 | 3 | The twisted arrow category of $\cal C$ is the category of elements of $\hom\_{\cal C}$.
>
> When is this category cofiltered?
>
>
>
This is equivalent to ask that the hom functor, taken as a presheaf on ${\cal C}^\text{op}\times \cal C$, commutes with finite limits, so
>
> When does $\hom\_{\cal C}$ commut... | https://mathoverflow.net/users/7952 | When is the twisted arrow category $\lambda$-cofiltered? | I think that it only happens in trivial cases. Let me assume that $\mathcal{C}$ has binary products and coproducts. The functor $\mathrm{Hom}\_{\mathcal{C}}$ preserves products if and only if for all $A,A',B,B' \in \mathcal{C}$ the canonical map
$$\mathrm{Hom}(A+A',B \times B') \longrightarrow \mathrm{Hom}(A,B) \times ... | 5 | https://mathoverflow.net/users/98306 | 251173 | 113,986 |
https://mathoverflow.net/questions/251177 | 1 | I'm looking for two papers written by H. Ishii:
* *Viscosity solutions of nonlinear partial differential equations*, Sugaku Expositions 9 (1996), no. 2, pp. 135--152 (English).
* *Viscosity solutions and their applications*, Sugaku Expositions 10 (1997), no. 2, 123--141 (English).
The library of my institution does... | https://mathoverflow.net/users/nan | Where can I find H. Ishii's expository papers that appeared on Sugaku Expositions (1996 and 1997)? | [Viscosity solutions and their applications](https://www.researchgate.net/publication/266365761_Viscosity_solutions_and_their_applications) is online via ResearchGate (and you can even read it without become a member, which is good). The other paper, [Viscosity solutions of nonlinear partial differential equations](htt... | 1 | https://mathoverflow.net/users/11260 | 251179 | 113,988 |
https://mathoverflow.net/questions/137059 | 16 | The $n\times n$ Hilbert matrix $H$ is defined as follows
$$H\_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$$
What is known about the singular values $\sigma\_1 \geq \cdots \geq \sigma\_n$ of $H$?
For example, it is known that the matrix is very ill-conditioned, i.e., [1]
$$\dfrac{\sigma\_1}{\sigma\_n} = \mat... | https://mathoverflow.net/users/2011 | The singular values of the Hilbert matrix | I came back to this a few months ago and I can now answer my own question. I hope it is appropriate to answer my own question given the length of time.
Bernhard Beckermann and I just submitted a paper [[1]](http://www.math.cornell.edu/~ajt/papers/PosDefHankel.pdf) that shows that if $AX-XB = F$ with $A$ and $B$ norm... | 8 | https://mathoverflow.net/users/2011 | 251187 | 113,991 |
https://mathoverflow.net/questions/251178 | 4 | An undirected graph may be regarded as a resistor network where each edge corresponds to a resistor of unit resistance. This paper covers such an approach.
* [*On electric resistances for distance-regular graphs*](https://arxiv.org/abs/1103.2810) by Jack Koolen, Greg Markowsky, and Jongyook Park
I paraphrase some s... | https://mathoverflow.net/users/41938 | reference request: voltage in a resistor network is a unique harmonic function | If I am reading correctly, this appears as Theorem 1.15 of Geoffrey Grimmett's book *Probability on Graphs*.
It is available for free on [Grimmett's web site](http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf) and has also been [published by Cambridge University Press](http://www.cambridge.org/us/academic/subj... | 5 | https://mathoverflow.net/users/4832 | 251188 | 113,992 |
https://mathoverflow.net/questions/251157 | 4 | Not related to [this](https://mathoverflow.net/questions/190513/model-bicategories) old question of mine, but takes the question from a different perspective.
Let $\mathcal V$ be a monoidal model category (following the def of Hovey, for example).
Then there is a bicategory $\text{Prof}(\mathcal V)$ of $\cal V$-value... | https://mathoverflow.net/users/7952 | The locally model bicategory of $\cal V$-profunctors | Given two model categories $\mathcal{M},\mathcal{N}$, one does know what would have been a left Quillen functor out of what would have been the tensor product $\mathcal{M} \otimes \mathcal{N}$ into a third model category $\mathcal{K}$, and that is a left Quillen bifunctor $\mathcal{M} \times \mathcal{N} \to \mathcal{K}... | 3 | https://mathoverflow.net/users/51164 | 251189 | 113,993 |
https://mathoverflow.net/questions/251192 | 7 | Is there a finitely presented sofic group which is not residually finite, but all of its finitely generated subgroups are Hopf groups?
It seems like the Baumslag Solitar groups $BS(m,n)$ don't work (i.e. for $|m|=1$ or $|n|=1$ or $|m|=|n|$ they are residually finite, and otherwise they contain a non-Hopf finitely gen... | https://mathoverflow.net/users/99167 | Non-residually-finite finitely-presented sofic group with all finitely generated subgroups Hopfian | Houghton's group $H\_3$ (see Section 5.3 [here](https://arxiv.org/abs/math/0511714) for a definition) is (locally finite)-by-$\mathbf{Z}^2$, ~~which easily implies that all its finitely generated subgroups are Hopfian~~. (Not all its subgroups are Hopfian: it admits an isomorphic copy of $F^{(\mathbf{N})}$ as a subgrou... | 8 | https://mathoverflow.net/users/14094 | 251202 | 113,995 |
https://mathoverflow.net/questions/251151 | 2 | I finally decided to post the following naive question but will if consensus is that it is out of the scope of this site , it will be immediately deleted.
Suppose $\Omega\subset\mathbb R^2$ is a bounded simply connected domain with smooth boundary and let $0<\lambda\_1^D(\Omega)\leq\lambda\_2^D(\Omega)\leq\cdots$ and... | https://mathoverflow.net/users/48438 | Universal constant for reverse inequality between first eigenvalues of Neumann and Dirichlet problems | No. Consider a thin rectangle, with side lengths $a\ll 1$ and $1$. Then $\lambda\_1^D=\pi^2(1+1/a^2)$ (eigenfunction $\sin \pi x\sin \pi y/a$), $\lambda\_2^N = \pi^2$ (eigenfunction $\cos \pi x$), and now you're in trouble when $a\to 0+$.
This domain does not have a smooth boundary, but of course you can iron out the... | 2 | https://mathoverflow.net/users/48839 | 251205 | 113,997 |
https://mathoverflow.net/questions/251161 | 5 | I realized my question [here](https://math.stackexchange.com/questions/1948807/burnside-group-b2-3-how-to-see-has-27-elements-and-isomorphic-to-certain) might have been too hard for MSE, so I'm asking it here as well.
The Burnside group $B(d, n)$ is defined as the quotient of the free group on $d$ generators by the n... | https://mathoverflow.net/users/99153 | Burnside group $B(2, 3)$ has $27$ elements, isomorphic to unitringular matrix group $\text{UT}(3, 3)$? | Since the question has been reopened, I mention a familiar argument which certainly appears in papers of G. Higman. If $G = \langle x,y \rangle$ is a non-Abelian group of exponent $3$, then since $xyxyxy = 1,$ we have $yxy = (xyx)^{-1} = x^{-1}y^{-1}x^{-1}.$ Hence $yx^{-1}x^{-1}y = x^{-1}yyx^{-1}$ as $x$ and $y$ both h... | 8 | https://mathoverflow.net/users/14450 | 251207 | 113,998 |
https://mathoverflow.net/questions/223837 | 3 |
>
> Let $G$ be a finitely generated residually free group.
>
>
>
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
>
> Let $H \lneq G$ be a proper finitely generated subgroup. Must there be
> a finite index proper subgroup... | https://mathoverflow.net/users/38889 | Is a finitely generated residually free group "almost LERF"? | *(converted from the comments)* No, $F\_2\times F\_2$ is a counterexample, where $F\_2$ is free on 2 generators.
Recall that a group is LPF if the profinite closure of every f.g. subgroup of infinite index has infinite index. This fails if there is a profinitely dense f.g. subgroup.
[Remark: your property appeared ... | 1 | https://mathoverflow.net/users/14094 | 251209 | 113,999 |
https://mathoverflow.net/questions/251191 | 3 | Let $A = [A\_1, \ldots, A\_m] \in \mathbb{R}^{n \times md}$, where for all $i=1,\ldots,m$, $A\_i \in \mathbb{R}^{n \times d}$, $d>1$. Let $x = [x\_1,\ldots,x\_m]^\top \in \mathbb{R}^m$ with $\|x\|\_2 \leq \varepsilon$, then what is a tight upper bound of $\big\| \sum\_{i=1}^m A\_i x\_i \big\|\_2$, i.e., the spectral no... | https://mathoverflow.net/users/97042 | spectral norm of block-wise sums of matrices | If I understand correctly, $\sum A\_i x\_i = A (x \otimes I\_d)$, so
$$
\|\sum A\_i x\_i\| \leq \|A\|\, \|x\otimes I\_d\| = \|A\|\, \|x\|,
$$
where the last inequality holds because $\|M\otimes N\|=\|M\|\,\|N\|$ for all matrices $M$, $N$ (which itself holds because the singular values of $M\otimes N$ are obtained by mu... | 6 | https://mathoverflow.net/users/1898 | 251210 | 114,000 |
https://mathoverflow.net/questions/251197 | 0 | Let $a>0,x\geq 0$, the lower regularized incomplete gamma function is defined as : $$P(a,x)=\frac{\gamma(a,x)}{\Gamma(a)} = \int\_0^x \frac{e^{-t}t^{a-1}}{\Gamma(a)}dt.$$
I have read in the paper of Gautschi "The incomplete gamma functions since Tricomi" that $P(.,x): a\mapsto \frac{\gamma(a,x)}{\Gamma(a)}$ is decrea... | https://mathoverflow.net/users/94576 | How to show $a\mapsto \frac{\gamma(a,x)}{\Gamma(a)}$ is decreasing on $\mathbb{R}_+^*$? | It is a little too elementary for this site, but here's the proof: First,
$$ \frac{\partial}{\partial a} P(a,x)=\frac{\Gamma(a)\partial\_a\gamma(a,x)-\gamma(a,x)\Gamma'(a)}{\Gamma(a)^2}. $$
So it suffices to show
$$\Gamma(a)\partial\_a\gamma(a,x)<\gamma(a,x)\Gamma'(a),$$
or
$$(\Gamma(a)-\gamma(a,x))\partial\_... | 4 | https://mathoverflow.net/users/37103 | 251213 | 114,002 |
https://mathoverflow.net/questions/251200 | 7 | Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(2)$ exists since $\mathfrak{M}(2)$ is smooth and proper). It is easy to see that $X(2)(\mathbb{C})$ is the modular curv... | https://mathoverflow.net/users/46460 | Modular curve X(2) | $y^2 =x (x-1)(x-\lambda)$ is a family of elliptic curves with level two structure on $\mathbb P^1\_{\mathbb Q}$. Hence by the definition of coarse moduli space, it defines a map from $\mathbb P^1$ to the coarse moduli space $X(2)$. It is sufficient to check that this map is an isomorphism and that it send $\lambda$ to ... | 6 | https://mathoverflow.net/users/18060 | 251217 | 114,003 |
https://mathoverflow.net/questions/251220 | 1 | This question is motivated by pedagogical reason, not research. I will provide a simple proof for contrast, but I would like to see another approach that does not involve integrals, instead even more elementary tools.
Prove that the sequence $a\_n$ converges if
$$a\_n=1+\sum\_{k=2}^n\frac1{k\log k}-\log\log n.$$
**Pr... | https://mathoverflow.net/users/66131 | An elementary proof for a limit? | As usually, you may replace integration by Lagrange mean value theorem. We have, denoting $f(x)=\log\log x$,
$$
a\_{n-1}-a\_{n}=(f(n)-f(n-1))-\frac1{n\log n}=f'(\theta\_n)-\frac1{n\log n}=\frac1{\theta\_n\log \theta\_n}-\frac1{n\log n},\\ n-1\leqslant \theta\_n\leqslant n.
$$
So, $a\_{n-1}-a\_{n}$ is positive, but the... | 3 | https://mathoverflow.net/users/4312 | 251221 | 114,004 |
https://mathoverflow.net/questions/250650 | 6 | Consider a torsion-free congruence subgroup $\Gamma\le SL\_2(\mathbb{Z})$ (for example, $\Gamma(N)$ for $N\ge 3$, or $\Gamma\_1(N)$ for $N\ge 4$).
By a meromorphic modular form for $\Gamma$ of weight $k$, I mean a holomorphic function $f$ on $\mathcal{H}$ with $f(\gamma z) = (cz+d)^kf(z)$ for $\gamma\in\Gamma$, which... | https://mathoverflow.net/users/88840 | Is the ring of meromorphic modular forms on a fine modular curve generated in degree 1? | Tyler basically answered the question in the comments, but I might as well fill in an answer. Your definition of "meromorphic modular form" is often called "weakly holomorphic modular form" in the literature, to distinguish such forms from those that may have poles in the interior of the half-plane.
Weakly holomorphi... | 3 | https://mathoverflow.net/users/121 | 251227 | 114,007 |
https://mathoverflow.net/questions/251226 | 0 | For any piece wise smooth, simple closed curve $\gamma$ in the Euclidean plane $E^2$ and fix a point $G$ inside the area circled by $\gamma$.
**Show**: There exists three points $A,B$ and $C$ on the $\gamma$, such that $G$ become the barycenter of the triangle $\Delta ABC$ and $\Delta ABC$ locates inside the area cir... | https://mathoverflow.net/users/95296 | Triangle inside the Closed Curve | This is true for convex curves, but false in general for non-convex ones.
Proof for convex curves: First find two points $A, D \in \gamma$ such that $G$ is the midpoint of $AD$ (by continuity argument or by intersecting $\gamma$ with its symmetral wrt $G$). Then take a point $E$ on $AD$ one third on the way from $D$ ... | 1 | https://mathoverflow.net/users/98590 | 251232 | 114,008 |
https://mathoverflow.net/questions/251233 | 6 | When is the Jacobian of a hyperelliptic curve
$$y^2=x(x-1)(x-a)(x-b)(x-c)$$
a product of two elliptic curves?
(This is a sort of reverse to
[When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?](https://mathoverflow.net/questions/35060/when-is-a-product-of-elliptic-curves-isogenous... | https://mathoverflow.net/users/9833 | When is the Jacobian a product? | A genus two curve $C$ has its jacobian isogenous to a product of elliptic curves if and only if there is a nonconstant map from $C$ to an elliptic curve $E$. For each degree of map $C \to E$ there is a hypersurface in the moduli space of genus two curves of such jacobians. So you have a countable union of hypersurfaces... | 5 | https://mathoverflow.net/users/1310 | 251237 | 114,009 |
https://mathoverflow.net/questions/225444 | 7 | Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r\_0)(x - r\_1)(x - r\_2)$; by Vieta's formulas, we know$$a = r\_0 + r\_1 + r\_2,\quad b = r\_0r\_1 + r\_1 r\_2 + r\_2r\_0,\quad c = r\_0r\_1r\_2.$$These expressions are invariant under three-cycles. Now, we make the... | https://mathoverflow.net/users/83593 | Precise relationship between "finite" Fourier analysis and Galois theory in solving the cubic? | The idea is that you can extract the explicit Kummer extension directly from the Fourier transform of the roots. If you have an element of the Galois group that cyclically permutes the roots, by weighting with appropriate roots of unity, i.e., taking a Fourier transform, the automorphism is diagonalized and acts on eac... | 6 | https://mathoverflow.net/users/121 | 251239 | 114,010 |
https://mathoverflow.net/questions/251225 | 5 | Let me first introduce the restricted setting in which this question has a nice answer. I came up with this when messing around with a homework problem in a PDE class a couple years back.
Let $\phi \in C^2[0,1]$. It induces a function $\phi\_\*: L^2[0,1] \to L^2[0,1]$, $f \mapsto \phi\circ f$. One can ask whether $\p... | https://mathoverflow.net/users/68932 | Which functions are continuous with respect to the weak topology? | Consider the sequence $f\_k:[0,1]^n\to\mathbb{R}$ defined by $f\_k(x\_1,\dots,x\_n):=(-1)^{\sum\_{j=1}^n \lfloor kx\_j \rfloor}$: it converges weakly to $0$.
For any $\phi:\mathbb{R}\to\mathbb{R}$, and for any $a$ and $b$ in $\mathbb{R}$, there holds: $$\phi\circ (a+bf\_k)=\frac{\phi(a+b)+\phi(a-b)}{2}+\frac{\phi(a+b)... | 3 | https://mathoverflow.net/users/6101 | 251245 | 114,012 |
https://mathoverflow.net/questions/251214 | 7 | That is to say, can one find a good bound on $|\pi\_i(S^n)|$? Let us assume that $i\ge 2n$ to avoid all infinite quantities. Particularly, I am interested to see if there is a bound of exponential type. I do not see a way to do this though.
| https://mathoverflow.net/users/41103 | Is there a bound on the growth of homotopy groups of spheres? | There are relevant estimates [in the 1986 paper by Hans-Werner Henn.](https://www.evernote.com/shard/s24/sh/f3bd9a0b-63d3-41ad-bf28-1473a8ee4cfd/a913c5482ab1fd26379e369f60403e59)
*Hans-Werner Henn*, MR 850372 [**On the growth of homotopy groups**](http://dx.doi.org/10.1007/BF01172158), *Manuscripta Math.* **56** (198... | 5 | https://mathoverflow.net/users/11142 | 251247 | 114,013 |
https://mathoverflow.net/questions/251259 | 3 | I am trying to understand the proof of Theorem 1 from [this](http://weisfeiler.com/boris/papers/1976-indag.pdf) paper V. Kac and B. Weisfeiler (Indag. Math. 1976, [DOI link](https://doi.org/10.1016/0003-4916(75)90068-8)).
>
> ***Theorem 1.** Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(Z^\maths... | https://mathoverflow.net/users/62601 | Harish-Chandra isomorphism for characteristic $p$ | In the previous lines it is proven that $U(T)^W$ is integral over $\gamma(Z^{\mathcal G})$. Since $U(T)^W\subset Frac(U(T)^W)=Frac(\gamma(Z^{\mathcal G}))$ the equality follows from $\gamma(Z^{\mathcal G})$ being integrally closed.
| 5 | https://mathoverflow.net/users/39304 | 251261 | 114,017 |
https://mathoverflow.net/questions/251250 | 4 | Let an algebraic group $G$ act on a complex variety $X$ such that there is a good enough quotient $X/G$ (for example, $G$ acts on a vector space $V$ linearly and $X=V\_{ss}$ is a variety of semi-stable points). Let $E$ be a vector bundle on $X$ with a $G$-action commuting with the $G$-action on $X$. I believe that if $... | https://mathoverflow.net/users/43639 | Vector bundles on quotient variety | I put my comment as an answer: the necessary and sufficient condition for $E$ to be the pull back of a vector bundle on $X/G$ is that the stabilizer of any closed point $x$ with a closed orbit
acts trivially on $E$. This is a lemma of Kempf, well explained in §2 of Drézet-Narasimhan *Groupe de Picard des variétés de ... | 8 | https://mathoverflow.net/users/40297 | 251265 | 114,019 |
https://mathoverflow.net/questions/251272 | 0 | Suppose you want an encoding of the integers $ \{0,1,2,\ldots,N-1\} $ into bitstrings. Two ways to do this are standard binary place value and unary encoding. The binary place value method is efficient in that it uses only $\log\_2 N$ bits. However, given an $x \in \{0,1,2,\ldots,N-1\}$, to check whether a bitstring en... | https://mathoverflow.net/users/23252 | encodings testable by querying few bits | Fix $k$ which will be your constant. Assume that the string length is $d$. Set $N={d\choose k}$, and to every $x\in \{0,\dots,N-1\}$ put into bijective correspondence a $k$-tuple of positions $F\_x$. Now you may encode $x$ by pussing ones eactly at the positions from $F\_x$, and then check `$x$-ness' by the same positi... | 1 | https://mathoverflow.net/users/17581 | 251273 | 114,022 |
https://mathoverflow.net/questions/251175 | 4 | Given $s\in (0,1)$ and a measurable function $u:\mathbb{R^n}\to\mathbb{C}$, let us define $$\|u\|\_{\dot H^s(\mathbb{R}^n)}^2:=\iint\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$
and let $\dot H^s(\mathbb{R}^n)$ denote the completion of $C^\infty\_c(\mathbb{R}^n)$ wrt this norm. One can define $H^s(\mathbb{R}^n)$ as the ... | https://mathoverflow.net/users/36952 | Homogeneous fractional Sobolev spaces | (1) No, not an equality. Look at the characterization in terms of Fourier transforms (\*).
(2) It depends on $s$. For $n=1$ and $\frac12\le s<1$ the completion is a quotient of the semi-Hilbert space defined by the seminorm being finite, with the subspace of constant functions. For all $n$ and $0<s<\frac{n}2$ it is a... | 6 | https://mathoverflow.net/users/75422 | 251277 | 114,023 |
https://mathoverflow.net/questions/251280 | 27 | It has been [estimated](http://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf) that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this original proof, with a targe... | https://mathoverflow.net/users/34444 | Has anyone catalogued the "first generation" proof of the classification of finite simple groups? | There are two books which together have the purpose of answering this question.
1. D. Gorenstein, The Classification of Finite Simple Groups. Volume 1: The Noncharacteristic 2 Type Case. Plenum Press, 1983.
(Gorenstein died without writing Volume 2.)
2. M. Aschbacher, R. Lyons, S.D. Smith, R. Solomon, The Classifica... | 42 | https://mathoverflow.net/users/99221 | 251282 | 114,024 |
https://mathoverflow.net/questions/251283 | 2 | Let $X$ be a fixed parametrizing space. Let $E$ and $E'$ be two spectra and let $E\_X$ and $E'\_X$ be their trivial parametrized versions. Intuitively I imagine that the morphisms of parametrized spectra $E\_X\to E'\_X$ should correspond to maps $X\to \operatorname{Map}(E, E')$ where the mapping space on the right shou... | https://mathoverflow.net/users/39713 | Are morphisms of parametrized spectra themselves parametrized morphisms of spectra? | This is true.
To prove it I will use the fact that maps of parametrized spectra can be computed as natural transformations of functors from $X$ into the $\infty$-category of spectra (cfr. [this paper](https://arxiv.org/abs/1403.4325)) and the formula for computing the space of natural transformation (e.g. see [here](... | 5 | https://mathoverflow.net/users/43054 | 251285 | 114,026 |
https://mathoverflow.net/questions/244567 | 12 | Let $p$ be a large prime and $n < p$. What is the smallest size of a set $A \subset \mathbb{Z} / p \mathbb{Z}$ such that $A \cdot \{1 , \ldots , n\} = \mathbb{Z} / p \mathbb{Z}$? Here $\cdot$ denotes the product set $X\cdot Y = \{xy : x \in X , y \in Y\}$.
Trivial bounds are $$p/n \leq |A| \leq p,$$ and using the pr... | https://mathoverflow.net/users/50426 | Small set such that $\{1 , \ldots , n\} \cdot A = \mathbb{Z} / p \mathbb{Z}$ | I believe that the answer to this question can be found in [this](https://arxiv.org/pdf/1310.0120v1.pdf) paper of Chen, Shparlinski and Winterhof. See Theorem 2 on page 6. It seems they give a contruction of a set with size $|A| \leq \frac{2p}{n}$ such that $A \cdot \{1,2,\dots,n\}=\mathbb Z\_p$.
| 8 | https://mathoverflow.net/users/23951 | 251305 | 114,032 |
https://mathoverflow.net/questions/251243 | 12 | The book *L'intégration dans les groupes topologiques et ses applications* published by André Weil in 1940 is regarded as one of the classical references for harmonic analysis on topological groups.
Unfortunately I am not fluent in French, so reading the book in all details is simply impossible. However, the reason ... | https://mathoverflow.net/users/nan | Weil's book L'intégration dans les groupes topologiques et ses applications | Leopoldo Nachbin's book "The Haar Integral" has Weil's proofs of existence and uniqueness of Haar measure, as well as Cartan's.
Weil establishes the Pontryagin duality theorem by an argument very similar to the original one by Pontryagin (in the compact/discrete case), which was extended to more general groups by van... | 7 | https://mathoverflow.net/users/99234 | 251307 | 114,034 |
https://mathoverflow.net/questions/251318 | 8 | The title is the question : Is there a non-metrizable topological space for which any countably compact subset is compact ?
EDIT : **non-metrizable and Hausdorff**
| https://mathoverflow.net/users/89425 | Is there a non-metrizable topological space for which any countably compact subset is compact? | Any Lindelöf non-metrizable Hausdorff space will do (EDIT: you need that the space is C-closed as well, see below), but more generally, a space is called isocompact iff every closed countably compact subset of X is compact, cl-isocompact iff the closure of a countably compact subset is compact, and C-closed iff any cou... | 7 | https://mathoverflow.net/users/29491 | 251322 | 114,037 |
https://mathoverflow.net/questions/251312 | 4 | A finite dimensional algebra A is called (n-)Igusa-Todorov in case there exists a module V such that for any module M there is an exact sequence:
$0 \rightarrow V\_2 \rightarrow V\_1 \rightarrow \Omega^{n}(M) \oplus P \rightarrow 0$, where $P$ is projective and $V\_i \in \mathrm{add}(V)$.
See [this 2009 paper by Jiaqun... | https://mathoverflow.net/users/61949 | Are all algebras Igusa-Todorov? | If $A$ is self-injective, then I think that $A$ being Igusa-Todorov implies that the dimension (in the sense of Rouquier) of the stable module category $\textrm{stmod-}A$ is at most $1$.
In "Dimensions of triangulated categories", J. K-theory 1 (2008), no.2, 193-256 ([link to arXiv](https://arxiv.org/abs/math/0310134... | 6 | https://mathoverflow.net/users/22989 | 251330 | 114,040 |
https://mathoverflow.net/questions/251341 | 1 | Let $\mathcal A \subset B(H)$ be an operator algebra and $\varphi: \mathcal A \rightarrow B(K)$ a completely bounded homomorphism. Suppose $\mathcal M \subset \mathcal A$ is an operator space such that $\overline{\rm Alg}(\mathcal M) = \mathcal A$.
>
>
> >
> > Does there exist a universal constant $C$ such that $... | https://mathoverflow.net/users/76593 | Norm of a cb-homomorphism restricted to a generating operator space | No, not even when $\cal A$ is commutative and $\phi $ is a multiplicative linear functional. For example, given $0<a<1$ let ${\cal A} =C[a,1]$, let $\cal M$ be the linear span of the function $f(x) = x$, and let $\phi $ be pointwise evaluation at $a$.
| 3 | https://mathoverflow.net/users/2554 | 251343 | 114,041 |
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