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https://mathoverflow.net/questions/336738 | 4 | Let $N,G,G^\prime$ be three affine algebraic group varieties (i.e geometrically reduced in the sense of J. Milnes) defined over a separably closed field $K$.
Suppose that we have the following exact sequence:
$$e\rightarrow N\rightarrow G\rightarrow G^\prime\rightarrow e$$
which means $N$ is a subgroup variety of $G$... | https://mathoverflow.net/users/143426 | k-points of an exact sequence of algebraic varieties | Yes, this is true. A group scheme over a field is smooth if and only if it is geometrically reduced, so the hypotheses ensure that $N$ is smooth. You can even allow $G$ and $G'$ to be arbitrary group schemes.
The map $G \rightarrow G'$ is an $N$-torsor, so it is a smooth morphism. This, for any $K$-point $g' \in G'(... | 7 | https://mathoverflow.net/users/56878 | 336775 | 143,776 |
https://mathoverflow.net/questions/234591 | 1 | I am trying to find out why the Busemann-Hausdorff area density as defined by [Burago and Ivanov](http://arxiv.org/pdf/1204.1543v2.pdf#page=3) is continuous. Here, $GC\_m(V)\subset \Lambda^m(V)$ denotes the simple $m$-vectors in an $n$-dimensional vector space $V$, that is, vectors $\sigma = v\_1 \wedge \cdots \wedge v... | https://mathoverflow.net/users/74130 | Continuity of Busemann-Hausdorff area density | Yes, it is continuous (and if the Finsler metric is smooth, it is also smooth). All you have to do to see it is to fix some Riemannian metric (or local coordinates) and verify that the volume of the unit tangent balls varies continuously (resp. smoothly) with the base point. The same goes for the Holmes-Thompson volume... | 2 | https://mathoverflow.net/users/21123 | 336782 | 143,779 |
https://mathoverflow.net/questions/336652 | 5 | While reviewing Lang's book on Arakelov theory, I saw the following comment by Paul Vojta:
"...Deligne has found an example when $\deg \pi\_{\*}\Omega\_{X/Y}$ can be negative, because Green's functions at infinity. This is of course unlike the functional field case, but this is of no consequence for next section..." ... | https://mathoverflow.net/users/18850 | Deligne's example of $\deg \pi_{*}\Omega_{X/Y}<0$ | This example should be the elliptic curve with j-invariant 0 and can be found in
Deligne's paper [Preuve des conjectures de Tate et de Shafarevitch](http://www.numdam.org/article/SB_1983-1984__26__25_0.pdf) (p. 29). If you are more interested in small values for elliptic curves, the article [On the essential minimum of... | 3 | https://mathoverflow.net/users/61532 | 336783 | 143,780 |
https://mathoverflow.net/questions/336786 | 3 | Suppose that $S$ is a perfect ring of char $p$ and is t-adically complete for some non-zero-divisor $t$. Is the ring of Witt vectors $W(S)$ $[t]$-adically complete? If so can i get a proof please?
| https://mathoverflow.net/users/143453 | Completeness of the ring of Witt vectors | This is actually true for any ring $S$ of characteristic $p$, and the argument I have in mind requires considering Witt vectors of non-perfect rings(such as $S/t^n$), so let me do it in full generality.
Fix an integer $k\geq 0$. For any ring $S$ with an ideal $I\subset S$ the canonical map induces an isomorphism $W\_... | 3 | https://mathoverflow.net/users/39304 | 336790 | 143,783 |
https://mathoverflow.net/questions/336788 | 1 | Let $s\in (0,1)$ and $u\in D^{s, 2} (\mathbb R^N)$, $\phi\in H^s\_{0}(B).$ Here $B$ is a unit ball in $\mathbb R^N$ and
$$H^s\_{0}(B)=\{ \phi\in H^{s} (\mathbb R^N): \phi=0 \text{ in } \mathbb R^N- B\}$$ and
$$D^{s, 2} (\mathbb R^N)= \bigg\{u\in L^{\frac{2N}{N-2s}}(\mathbb R^N): \int\_{\mathbb R^{2N}}\frac{|u(x)-u(y)... | https://mathoverflow.net/users/139853 | Integration by parts in nonlocal case | It looks like the definition of $D^{s,2}(\mathbb{R}^N)$ is too weak for $(-\Delta)^s u$ to be defined in the usual way. (One could possibly move to distributional definitions, in the sense of $H^{-s}$ or similar objects, but then the desired formula seems would sort of take us back to the definition of $H^{-s}$, so I l... | 1 | https://mathoverflow.net/users/108637 | 336792 | 143,784 |
https://mathoverflow.net/questions/331217 | 2 | If we want to certify the nonexistence of real solutions to a polynomial system of equations, i.e.
$$ S = \{ x\in \mathbb{R}^n \ | \ h\_i (x) = 0, \ i=1,\dots,t \} = \emptyset, $$
we can produce a Positivstellensatz refutation by writing $-1$ as a Sum of Squares (SOS) polynomial $s$ modulo the ideal generated by th... | https://mathoverflow.net/users/140486 | SDP representation of ideal polynomials for positivstellensatz refutations | As I guessed, I was missing a fairly trivial piece (and most of my question is not even relevant to the answer).
Free decision variables $a\_i$ can be represented in SDPs by the difference of two positive variables $x\_{i,1} - x\_{i,2}$, and positive variables $x\_{i,j}$ can be incorporated by enlarging the positive ... | 0 | https://mathoverflow.net/users/140486 | 336800 | 143,787 |
https://mathoverflow.net/questions/336793 | 6 | A classical result of simply connected surgery theory, is that if two normal maps $f:M\_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a homotopy sphere $\Sigma$ such that $M\_0$ is diffeomorphic to $M\_1\#\Sigma$. This was first proved by Novikov in
*Homot... | https://mathoverflow.net/users/142933 | Diffeomorphism type of the added sphere in simply connected surgery | The ambiguity in the the added homotopy sphere is captured by a suitable *inertia group $I(X)$*, which in this case is the group of homotopy spheres $\Sigma$ such that $\Sigma$ bounds a parallelizable manifold, and the standard homeomorphism $\Sigma\# X\to X$ is homotopic to the diffeomorphism.
There is a related ine... | 5 | https://mathoverflow.net/users/1573 | 336804 | 143,788 |
https://mathoverflow.net/questions/336771 | 11 | Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it implies that only finitely many torsion subgroups are possible, which seems like a much weaker result.
My question: Is there... | https://mathoverflow.net/users/140821 | Bounded Torsion, without Mazur’s Theorem | By all means hold out hope, but I don't think that the ideas in Dem'janenko's papers are going to work. I spent a lot of time in grad school looking at them.
If I remember correctly, Dem'janenko also claimed to have proven that on $E:y^2=x^3+D$, if a $P\in E(\mathbb Q)$ is a non-torsion point, then $\hat h(P)\ge c\l... | 16 | https://mathoverflow.net/users/11926 | 336808 | 143,791 |
https://mathoverflow.net/questions/336803 | 7 | Consider a manifold $M$. Then, we have the notion of differential forms on $M$ and complex associated to that, denoted by $$\cdots\rightarrow \Omega^{k-1}(M)\rightarrow \Omega^k(M)\rightarrow \Omega^{k+1}(M)\rightarrow \cdots$$
giving de Rham cohomology groups $H^k\_{\mathrm{dR}}(M)$ for $k\in \mathbb{N}$.
Now consid... | https://mathoverflow.net/users/118688 | Differential forms of a Lie group giving cohomology of the Lie group | For every Lie group $G$, you have in your $\Omega\_G$ the subcomplex of the left-invariant differential forms. Moreover, $G$ acts on the right on this subcomplex. It has the virtue of being finite-dimensional and isomorphic to a complex defined purely in terms of the Lie algebra, thus the computation of its cohomology ... | 7 | https://mathoverflow.net/users/105095 | 336810 | 143,793 |
https://mathoverflow.net/questions/336812 | 1 | Let $X$ be a separable pointed metric space and let $AE(X)$ denote the corresponding Lipschitz-Free (or Arens-Eells space) over $X$. The point-evaluation map $\delta:X\mapsto AE(X)$ is injective hence it has a left-inverse. When $X$ is Banach, the left-inverse may be taken to be continuous and linear.
In general, if... | https://mathoverflow.net/users/36886 | Continuous Left-inverse of Dirac Lipschitz-Free Space | You're going to need that $X$ is contractible at least.
Let $X$ be the unit circle with the metric inherited from $\mathbb{R}^2$ and some point chosen as a base point. Assume that $f$ is a retraction of $AE(X)$ onto $X$. Since $AE(X)$ is a Banach space it is contractible. Let $g:AE(X)\times[0,1]\rightarrow AE(X)$ be ... | 4 | https://mathoverflow.net/users/83901 | 336814 | 143,796 |
https://mathoverflow.net/questions/336781 | 6 | I have to deal with unbounded filtrations and want to use the conditional convergence of spectral sequences and the results from
[1]: *J. Michael Boardman, Conditionally Convergent Spectral Sequences, March 1999 (<http://hopf.math.purdue.edu/Boardman/ccspseq.pdf>)*
The article uses cohomological spectral sequences... | https://mathoverflow.net/users/43645 | Conditionally convergent spectral sequences with exiting and entering differentials | After a long life in preprint form, Boardman's paper was published in the conference proceedings celebrating his 60th birthday:
```
\bib{MR1718076}{article}{
author={Boardman, J. Michael},
title={Conditionally convergent spectral sequences},
conference={
title={Homotopy invariant alg... | 6 | https://mathoverflow.net/users/9684 | 336815 | 143,797 |
https://mathoverflow.net/questions/335990 | 2 | Suppose that we have a sequence $i: X \hookrightarrow Y$, $j: Y \hookrightarrow Z$ of closed embeddings of varieties such that $i$ is regular. In this case, do we have an exact sequence of cones of the form
$N\_{X}Y \rightarrow C\_{X}Z \rightarrow i^{\*} C\_{Y}Z$,
where $i^{\*} C\_{Y}Z := C\_{Y}Z \times \_{Y} X$? ... | https://mathoverflow.net/users/16981 | Exact sequence of normal cones | This is false, a sequence like that need not necessarily be exact.
The counterexample is the same as in Fulton, with $Z$ a cone over a quadric, $Y$ a line through the vertex and $X$ the vertex itself; see [Virtual Classes for the Working Mathematician](https://arxiv.org/pdf/1804.06048.pdf), 3.5.1.b).
| 1 | https://mathoverflow.net/users/16981 | 336816 | 143,798 |
https://mathoverflow.net/questions/336625 | 1 | Consider the measurable space $(\Omega,\mathscr{B})$ endowed with two positive measures: a "volume $\nu$" and a probability measure $\mu$. For example, one might take $\Omega=\mathbb{R}^n$ (with the usual $\sigma$-algebra) and $\nu$ as the Lebesgue measure.
I would like to quantify the notion of "a high proportion of... | https://mathoverflow.net/users/12518 | (Novel?) notion of concentration/dispersion | The function you propose is related to the L'evy concentration function,
studied by Kolmogorov, Rogozin, Esseen
and others. See the special volume [1] <https://link.springer.com/chapter/10.1007/978-94-011-2260-3_70>
The classic book [2] has a chapter devoted to concentration functions with many references and the p... | 2 | https://mathoverflow.net/users/7691 | 336821 | 143,800 |
https://mathoverflow.net/questions/336827 | 5 | Let $K$ be a compact Hausdorff infinite topological space and $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$ with sup norm. It is known that $c\_0$ is complemented in $C(K)$ for $K$ that contains infinite convergent sequence. I would appreciate if somebody let me know an example of $K$ without... | https://mathoverflow.net/users/143479 | Compact space $K$ without convergent sequence while $c_0$ is complemented in $C(K)$ | Take $K = \beta N \times \beta N$, the square of the Čech--Stone compactification of the integers. The space of continuous functions on that space is not a Grothendieck spaces. A space of continuous functions on a compact space is Grothendieck if and only if it doesn't contain complemented copies of $c\_0$.
More gene... | 3 | https://mathoverflow.net/users/15129 | 336830 | 143,802 |
https://mathoverflow.net/questions/336797 | 4 | Given a representation-finite (connected) quiver algebra $A$ and a module $M$.
>
> Is there a good way to test whether the set $\{ [N] \mid N \in \mathrm{add}(M) \}$ generates $K\_0(\mbox{mod-}A)$?
>
>
> Can this be done using the GAP-package QPA?
>
>
>
| https://mathoverflow.net/users/61949 | Testing whether a module generates $K_0(\mbox{mod-}A)$ | Suppose that $M = \oplus\_{i=1}^n M\_i$ with $M\_i$ being indecomposable and assume that $M\_i \not\simeq M\_j$ for $i\neq j$ (that is, $M$ is basic) over a finite dimensional algebra $A$. Define a matrix $K$ with rows equal to the dimension vector of $M\_i$ for $i = 1,2,\ldots,n$. Then $M$ generates Grothendieck group... | 4 | https://mathoverflow.net/users/130741 | 336847 | 143,809 |
https://mathoverflow.net/questions/336829 | 1 | Let $A\_i, i=1, \ldots, N$ be real (or complex) matrices of the same dimension. Let $r\_i, i=1, \ldots, N$ be independent Rademacher random variables.
The following inequality gives a bound on the expectation of the p-th Schatten norm:
$$
E\left\|\sum\_{i=1}^N r\_i A\_i\right\|\_{S\_{p}}^{p}\leq C\sqrt p \max\left\{\l... | https://mathoverflow.net/users/122182 | Matrix inequalities for the moment of the fixed Shatten norm | If you do not care about the precise constants, the answer is yes, and this follows directely from the Kahane inequality (which says that, in any normed space and $1 \leq p,q<\infty$, the $L\_p$-norm of $\sum r\_i x\_i$ is dominated by its $L\_q$-norm, up to constant depending on $p,q$ only). The conclusion is that for... | 1 | https://mathoverflow.net/users/10265 | 336850 | 143,810 |
https://mathoverflow.net/questions/336860 | 4 | Let $S$ be a closed hyperbolic surface of genus $g\geq 2$. Let $(\mathcal{T},\omega)$ be the corresponding Teichmuller space with the Weil–Petersson symplectic from $\omega$. Let $\Phi:\mathcal{T}\rightarrow\mathcal{T}$ be any diffeomorphism which preserves $\omega$.
**Q) Does there exist a diffeomorphism $\phi:S\rig... | https://mathoverflow.net/users/37286 | Are symplectomorphisms of Weil–Petersson symplectic form induced from surface diffeomorphisms? | There are infinitely many compactly supported symplectomorphisms of any symplectic manifold, which would then have to be represented by diffeomorphisms of $S$ preserving all marked conformal structures, except those in the compact set. But preserving such a conformal structure is only possible for a finite set of diffe... | 7 | https://mathoverflow.net/users/13268 | 336861 | 143,814 |
https://mathoverflow.net/questions/336856 | 3 | The following concentration inequality for the supremum of a Gaussian process indexed by a separable metric space appears here: <http://math.iisc.ac.in/~manju/GP/6-Concentration%20and%20comparison%20again.pdf>
(this, to the best of my knowledge, is one of several lecture notes prepared by Prof. Manjunath Krishnapur, II... | https://mathoverflow.net/users/143501 | Can anyone give a reference to the proof of this concentration inequality? | This follows immediately from the [Borell-TIS inequality](https://en.wikipedia.org/wiki/Borell%E2%80%93TIS_inequality).
Indeed, for any $x>EX^\*$ this inequality yields
$$\frac1{x^2}\,\ln P(X^\*\ge x)\le-\frac{(x-EX^\*)^2}{2\sigma\_T^2 x^2}\to-\frac1{2\sigma\_T^2}
$$
as $x\to\infty$.
On the other hand, for any $... | 1 | https://mathoverflow.net/users/36721 | 336872 | 143,821 |
https://mathoverflow.net/questions/336864 | 7 | Could you please let me know if the following is true. The problem came up while constructing a solution of a PDE. I have browsed through the net for an answer. While I came across some articles regarding the identity component of the diffeomorphism group, with my poor geometry and topology I could not really figure ou... | https://mathoverflow.net/users/27832 | Path of Diffeomorphisms Fixing the Boundary | A diffeomorphism for which there exists such an H is called an isotopy (relative to the boundary).
This has been much studied in the case where Omega is the unit ball in R^n. Your question is well-known as: does pi\_0(D^n,\partial D^n)=0?
The answer is positive for the unit balls in R^2 (Smale 1958) and R^3 (Cerf'... | 13 | https://mathoverflow.net/users/105095 | 336873 | 143,822 |
https://mathoverflow.net/questions/336866 | 3 | Let $L$ be a first order language and $M$ an interpretation of $L$.
If $A$ and $B$ are two substructures of $M$ and their intersection $C=A\cap B\ne \emptyset$, then is it the case that every sentence (in language $L$) true for both $A$ and $B$ is also true for $C$?
By "intersection" of $A$ and $B$, I mean the smalle... | https://mathoverflow.net/users/143420 | Is a sentence true for two substructures also true for their intersection? | If $\mathcal C$ is any substructure of a structure $\mathcal A$, then $\mathcal C$ is the intersection of $\mathcal A$ and an isomorphic copy of it, $\mathcal A'$, inside a larger structure $\mathcal M$. (Take an isomorphic copy of $\mathcal A$ and combine it with $\mathcal A$, identifying corresponding elements of $\m... | 10 | https://mathoverflow.net/users/6794 | 336877 | 143,825 |
https://mathoverflow.net/questions/336884 | 7 | The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes\_{\mathcal{A}\_2} \mathbb{F}\_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $\mathcal{A}\_2$ denotes the subalgebra generated by $Sq^1$ and $Sq^2$.
Where can I find the original calculation for referencing ... | https://mathoverflow.net/users/21985 | Reference request: mod 2 cohomology of periodic KO theory | Ravenel in his *Complex Cobordism and Stable Homotopy Groups of Spheres* attributes this result to Stong, in [*Determination of $H^\*(BO(k,⋯,∞),Z\_2)$ and $H^∗(BU(k,⋯,∞),Z\_2)$*](https://www.ams.org/journals/tran/1963-107-03/S0002-9947-1963-0151963-5/S0002-9947-1963-0151963-5.pdf), but looking at that paper (which is c... | 10 | https://mathoverflow.net/users/43054 | 336886 | 143,827 |
https://mathoverflow.net/questions/336824 | 2 | In this post I invoke certain function from a post of this site MathOverflow it is [1] (please see further references from the post, authors from the Springer link of the cited literature and answers from [1]: the integral mentioned in the post [1] seems thus that is due Lobachevskii, *Application of imaginary geometry... | https://mathoverflow.net/users/142929 | On $\zeta(7)$ as the integration of the product of an indefinite integral due to Lobachevskii by a power of the inverse Gudermannian function | First, $\mathrm{gd}^{-1}(z) = \ln \frac{1 + \tan(\frac{z}{2})}{1 + \tan(\frac{z}{2})}$. So we substitute $t = \tan(\frac{z}{2})$, obtaining:
$$\mathcal{G}\_n = \int\limits\_{0}^{1}2\cdot\ln^{n + 1}\left(\frac{1 + t}{1 - t} \right)\frac{d t}{t}$$
Then take $\frac{1+t}{1 - t} = e^{x}$, to get:
$$ \int\limits\_{0}^{\inf... | 18 | https://mathoverflow.net/users/141327 | 336887 | 143,828 |
https://mathoverflow.net/questions/336852 | 4 | Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with with respect to coproducts as well (that it, it is an AB3 abelian category).
Indeed, the aisle $T^{t\le 0}$ is closed with... | https://mathoverflow.net/users/2191 | Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)? | This is Proposition 3.1.2 in the [thesis of Parra](https://arxiv.org/abs/1409.6639), "Hearts of $t $-structures which are Grothendieck or module categories".
| 4 | https://mathoverflow.net/users/22989 | 336899 | 143,831 |
https://mathoverflow.net/questions/336895 | 2 | We are reading a proof about the following limit
\begin{equation}\tag{1}
\lim\_{n \to \infty} \sigma\_1(T\_n)= \sigma\_1(T),
\end{equation}
where $T:D(T) \subseteq H \to H$ and $T\_n:D(T\_n) \subseteq H \to H$ are linear operators on a Hilbert space $H$ and
$$\sigma\_1(A)= \sigma(A) \cup \{ z \in \mathbb{C}: \|(z-A)^{-... | https://mathoverflow.net/users/142048 | Understanding a proof about limit of a sequence of open sets | You have hit the nail on the head and indeed there is a mistake with the proof you are reading. The argument already fails for matrices. If $T\_n \equiv T \in \mathbb{C}^{n \times n}$ then $\sigma\_1(T)$ is open by definition, but
$$ \lim \sigma\_1(T\_n) = \overline{\sigma\_1(T)}, $$
as any element in the closure of $... | 5 | https://mathoverflow.net/users/85570 | 336900 | 143,832 |
https://mathoverflow.net/questions/336892 | 11 | Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and
$$\int\_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for every $1\leq p\leq 2$?
| https://mathoverflow.net/users/117091 | Function orthogonal to powers of $1/\left(1+x^2\right)$ | I think no such function exists. Assuming $\displaystyle{f(x)\over 1+x^2}\in L^1(\mathbb{R})$ the integral $\displaystyle\int\_\mathbb{R}{f(x)\over (1+x^2)^p}dx$ is analytic wrto $p>1$: indeed, for any $p>1$ and $|t|<p-1$, expanding $(1+x^2)^{t}$ in powers of $t$ we have, by Tonelli's theorem (w.rto the product measure... | 13 | https://mathoverflow.net/users/6101 | 336909 | 143,833 |
https://mathoverflow.net/questions/336905 | 4 | Let $X$ be a closed symplectic manifold equipped with a smooth Lagrangian torus fibration $\pi:X \rightarrow Q$. Assume that $\pi$ admits a Lagrangian section. By work of Kontsevich-Soibelman, one can associate to $(X, \pi)$ a rigid analytic (in the sense of Tate) space $Y$. In recent works, Abouzaid showed that one ca... | https://mathoverflow.net/users/123002 | Mirror symmetry for singular Lagrangian torus fibrations | Yes, there is a concrete program on how to handle singular fibers in the SYZ fibration and several steps of this program are already completed.
You can watch the videos of Abouzaid lectures on the subject at
<https://www.youtube.com/watch?v=1PqIE3YJU0I>
<https://www.simonsfoundation.org/event/simons-collaboration... | 5 | https://mathoverflow.net/users/439 | 336936 | 143,838 |
https://mathoverflow.net/questions/336933 | 3 | Let $A$ be a ring, $S$ the spectrum of $A$, $f : E \to S$ an elliptic curve.
Then assuming $f\_\*\Omega\_{E/A}$ is free over $S$, $\hat{E}$ (the formal completion along the $0$-section.) $ \cong \operatorname{Spf}(A[[T]])$.
I intuitively think that this isomophism means "$E$ is, locally at $0$, just the line",
and th... | https://mathoverflow.net/users/128235 | Formal group and formal completion of an elliptic curve | On questions 3-5:
3. Yes, EGA I, new edition, comment after 10.2.2.
4. Yes, EGA I, new edition, Proposition 10.7.2.
5. This follows fro the fact that F is the comultiplication of a coalgebra, in particular the image of T is not invertible.
I guess your 1 and 2 follow from the description of a formal scheme as a cer... | 4 | https://mathoverflow.net/users/6348 | 336941 | 143,840 |
https://mathoverflow.net/questions/336928 | 2 | The Wikipedia's article for *Prime number* shows a known and curious formula for primes from its section [*Formula for primes*](https://en.wikipedia.org/wiki/Prime_number#Formulas_for_primes), I say the Mills' theorem (please see also the Wikipedia *Mills' constant*).
>
> **Question.** I wondered if one can to dete... | https://mathoverflow.net/users/142929 | What about a formula similar than Mill's formula, but producing positive integers without repeated prime factors? | Along the lines of the [Wikipedia page](https://en.wikipedia.org/wiki/Square-free_integer#Distribution), it is true that
$|Q(x)-\frac{x}{\zeta(2)}|\leq2+\sqrt{x}$
where $Q(x)$ is the number of square-free numbers between $1$ and $x$.
So,
$Q(n^3)\leq\frac{n^3}{\zeta(2)}+2+n^\frac{3}{2}$
$Q((n+1)^3)\geq\frac{(... | 5 | https://mathoverflow.net/users/125498 | 336942 | 143,841 |
https://mathoverflow.net/questions/336937 | 6 | I am wondering if the 2-сategory of groupoids is locally presentable. Locally presentable means the category is accessible and co-complete.
It has been pointed out that the category of groupoids is locally presentable. It would be useful if I could find a theorem that says that if your base category, namely groupoids... | https://mathoverflow.net/users/10007 | Is the 2-сategory of groupoids locally presentable? | This is true.
Since the $2$-category of groupoids is equivalent to the $2$-category of $1$-truncated spaces, the statement follows immediately from 5.5.1.8 and 5.5.6.21 in Lurie's Higher Topos Theory.
| 8 | https://mathoverflow.net/users/16981 | 336950 | 143,844 |
https://mathoverflow.net/questions/336949 | 2 | Let $X\_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring
$$S = \mathbb{C}[x\_{\rho}\: | \: \rho\in\Sigma(1)]$$
Define $\deg(x\_{\rho}) = D\_{\rho}$.
Now, take a divisor $D = \sum\_{\rho\in\Sigma(1)}a\_{\rho}D\_{\rho}$. Then $H^0(X,D)$ is generated by monomials $X^{m}$ such that $m\in M$ with ... | https://mathoverflow.net/users/nan | Sections of Cartier divisors on toric varieties | That is true. Just note that
$$deg(x\_1^{<m,u\_1>+a\_1}\cdots x\_k^{<m,u\_k>+a\_k})$$
is the class in $Cl(X\_{\Sigma})$ of the divisor
$$\sum\_{\rho} (<m,u\_\rho>+a\_\rho)D\_\rho = \sum\_{\rho}<m,u\_\rho>D\_\rho + \sum\_{\rho}a\_{\rho} D\_\rho \sim div(\chi^m)+D\sim D$$
| 2 | https://mathoverflow.net/users/14514 | 336957 | 143,847 |
https://mathoverflow.net/questions/336958 | 4 | I am reconsidering an argument previously I thought obvious but now I feel I do not understand. Let $G$ be a finite flat group scheme over a finite field $k$. $G^\vee$ the Cartier dual of it. Let $TG$ denote its Lie algebra. $\mathbb{G}\_a$ the additive group scheme. Then we have
$$
TG\cong \mathrm{Hom}(G^\vee, \math... | https://mathoverflow.net/users/98747 | Tangent space of a finite flat group scheme | This is correct. Here's another to see it (assume for simplicity we are working over a field $k$). Let $R\_\epsilon : = k[\epsilon]/\epsilon^2$ and let $S\_\epsilon:= Spec R\_\epsilon.$ This is a scheme with a single $k$-point, $s\in S$ and a one-dimensional tangent space. Now recall that the tangent space can be inter... | 5 | https://mathoverflow.net/users/7108 | 336966 | 143,848 |
https://mathoverflow.net/questions/336968 | 13 | I asked this on mathstackexchange, but got no answer.
Let $N\geq1$ be an integer, and let $\mathbb{T}$ be the Hecke algebra acting on the cusp forms of weight k and level $\Gamma\_0(N)$.
Then $\mathbb{T}$ is finite and flat over $\mathbb{Z}$, but the proof I know for this fact is slightly roundabout, by letting it... | https://mathoverflow.net/users/143589 | How does one compute the Hecke algebra acting on modular forms? | This isn't quite an answer, but since I cant comment, I'll do it here.
In MAGMA you can ask for HeckeAlgebra of a space of ModularSymbols (or maybe it only works for the cuspidal subspace of such), but in any case, it'll give you some generators and such. You can see more here <https://magma.maths.usyd.edu.au/magma/h... | 3 | https://mathoverflow.net/users/143607 | 336982 | 143,854 |
https://mathoverflow.net/questions/336981 | -1 | If $G=(V,E)$ is a simple, undirected graph, we say that $M\subseteq E$ is a *perfect matching* if the members of $M$ are pairwise disjoint and $\bigcup M = V$. Let $K\_\omega$ be the complete graph on $\omega$.
Let ${\frak M}$ be a class of perfect matchings on $K\_\omega$ such that whenever $M\_1\neq M\_2\in{\frak ... | https://mathoverflow.net/users/8628 | Cardinality of a set of mutually disjoint perfect matchings of $K_\omega$ | It's reasonably clear that there can be at most countably many disjoint perfect matchings, since each matching must contain an edge adjacent to each vertex and there are only countably many such edges.
Conversely it's not to hard to build a countable family of perfect matchings in the following manner:
Our aim will... | 3 | https://mathoverflow.net/users/35545 | 336987 | 143,856 |
https://mathoverflow.net/questions/336988 | 2 | Noah Schweber said [here](https://math.stackexchange.com/questions/1902347/grothendieck-topology-and-relation-with-usual-topologies) the following:
>
> Why would you want a notion of sheaf theory for objects more general
> than topological spaces? Well, the original motivation (to my
> understanding) was to devel... | https://mathoverflow.net/users/118688 | Why care about Grothendieck topology? | Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only.
In the category of topological manifolds, an etale cover of $X$ is a surjective local homeomorphism $Y\rightarrow X$. In the etale topology on varieties, this is "morally true" as well with loc... | 10 | https://mathoverflow.net/users/5301 | 336991 | 143,858 |
https://mathoverflow.net/questions/337004 | 3 | Here is the notation. Let $k$ be a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vector and $\mathfrak{S}:=W[[u]]$, $\mathcal{O}\_{\mathcal{E}}$ is the $p$-adic completion of $W[1/u]$ which is DVR. Denote $R=\lim\_{\leftarrow}\mathcal{O}\_{
\bar{K}}/p$ where the transition map is Frobenius. The ring of... | https://mathoverflow.net/users/143571 | maximal unramified extension of Breuil ring in $A_{cris}$ | It is false that $\mathfrak{S}^{un}\cap pA\_{cris}=p\mathfrak{S}^{un}$. For example, $E(u)\in\mathfrak{S}\subset\mathfrak{S}^{un}$ is not divisible by $p$ in $\mathfrak{S}^{un}$ but gets mapped to $\varphi(\xi)$ where $\xi$ is a generator of $\theta:W(R)\to\mathcal{O}\_C$ which is divisible by $p$ in $A\_{cris}$ becaus... | 3 | https://mathoverflow.net/users/39304 | 337018 | 143,863 |
https://mathoverflow.net/questions/335371 | 2 | Let $k$ be an integer such that $k>n/2$, and let $H^k(\mathbb{R}^n)$ denote the usual Sobolev Hilbert space.
Let $f,g\in H^k(\mathbb{R}^n)$.
Is it true that
$\displaystyle
\lim\_{R\rightarrow \infty} \int\_{S\_R} f\cdot g\; dS\_R=0?
$
Here $S\_R$ denotes the sphere of radius $R$ centered at $0$, and $dS\_R$
... | https://mathoverflow.net/users/142650 | On the limiting behaviour of Sobolev space functions | If $k>n/2$ then you have (continuously) $H^k(\mathbf{R}^d)\hookrightarrow L^\infty(\mathbf{R}^n)$ by Morrey's Theorem. Since compactly supported functions are dense in $H^k(\mathbf{R}^d)$ you get by uniform convergence (thanks to the previous embedding) that elements of $H^k(\mathbf{R}^d)$ tend to $0$ at infinity.
| 2 | https://mathoverflow.net/users/27767 | 337028 | 143,867 |
https://mathoverflow.net/questions/337036 | 1 | I was interested in create and solve a Diophantine equation similar than was proposed in the section **D3** of [1]. I would like to know what theorems or
techniques can be applied to prove or refute that the Diophantine equation of the title has a finite number of solutions, I don't have the intuition to know it. Our ... | https://mathoverflow.net/users/142929 | Solutions of $y^2=\binom{x}{0}+2\binom{x}{1}+4\binom{x}{2}+8\binom{x}{3}$ for positive integers $x$ and $y$ | Start with
$$ 3^2=4^3−6^2+8+3. $$
The change of variables $x=X/12$ and $y=Y/36$ gives the equation
$$ Y^2 = X^3 - 18X^2 + 288X + 1296. $$
Entering this into the LMFDB leads to the page
<http://www.lmfdb.org/EllipticCurve/Q/315936/g/1> .
So your elliptic curve, after another change of variables to get rid of the $X^2$ t... | 11 | https://mathoverflow.net/users/11926 | 337046 | 143,875 |
https://mathoverflow.net/questions/337021 | 1 | **Questions:**
assuming
$$a\lt b,\ c\lt d;\ \ (x,y)\in [a,b]\times[c,d];\ \ f\_0: (x,y)\mapsto z\in\mathbb{R};\ \ |a|,\ |b|,\ |c|,\ |d|,\ |z|\lt\infty$$
$$0\quad\lt\quad\left|\int\_a^b{f\_0(x,y)dx}\right|,\ \ \left|\int\_c^d{f\_0(x,y)dy}\right|$$
$$f\_{i+1}(x,y) := f\_i(x,y)-\frac{(d-c)\int\_a^b{f\_i(t,y)dt}\ +\... | https://mathoverflow.net/users/31310 | Effect of repeated subtraction of the average of average function values in coordinate directions | Denote $I:=[a,b]$, $J:=[c,d]$ and say $f$ is in $L\_2(I\times J)$. You are iterating the bounded linear operator ${\bf1}-{1\over2}(P+Q)$, where $Pf(x,y):={1\over|I|}\int\_If(s,y)ds$ and $Qf(x,y):={1\over|J|}\int\_Jf(x,t)dt$ take the mean of $f$ on the first, resp. second variable; that is, $P$ and $Q$ are the linear (o... | 1 | https://mathoverflow.net/users/6101 | 337056 | 143,880 |
https://mathoverflow.net/questions/337068 | -1 | Consider a simple regular graph on $n$ vertices and size $E$. How many distinct maximal independents can we find at the least in the graph?
I think we can always find at least two maximal independent sets in the graph. This is an inspiration from the observation on bipartite graphs. In case of multipartite sets, the... | https://mathoverflow.net/users/100231 | Number of maximal independent sets in a simple graph | Assume $G$ has chromatic number $χ$.
Let $a\_1,a\_2,...,a\_χ$ be a partition of $G$ into $χ$ independent sets (This is always possible given the chromatic number).
Every set $a\_k$ can be extended to a maximal independent set $A\_k$. The $A\_k$s are pairwise different. Otherwise, assume $A\_p=A\_q$, it follows that $... | 4 | https://mathoverflow.net/users/125498 | 337071 | 143,887 |
https://mathoverflow.net/questions/337080 | 1 | What is the maximal cardinality of a subset $A$ of $\{-1,1\}^n$ such that any Hamming sphere with radius $r$ contains at most $k$ elements of $A$?
Are explicit constructions with large cardinality known?
Any relevant comment or reference appreciated.
| https://mathoverflow.net/users/112954 | Large subsets of the Hamming cube with small intersections with all spheres of given radius | Two words: list decoding
Apparently the previous answer is too short to be an answer, so I am writing this sentence.
| 3 | https://mathoverflow.net/users/143679 | 337104 | 143,899 |
https://mathoverflow.net/questions/337106 | 0 | Is there a standard (or at least common) symbol in computability theory used to indicate that $x$ enters the c.e. set $W\_e$ at stage $s$, i.e., $x \in W\_{e,s} - W\_{e,s-1}$ (at least for $s \neq 0$)?
I was thinking of using $x \searrow\_s W\_e$ but I was worried that would be too confusing with its use in the auto... | https://mathoverflow.net/users/23648 | Computability Theory Notation For Entering A Set At A Stage | (I'm going to write "$V$" for the arbitrary c.e. set, since I'm rather wedded to the convention that "$W\_e$" refers to the domain of the $e$th partial computable function.)
---
The "$\searrow\_s$" notationis what I've seen before - I think it's in Soare's old book, although I don't have my copy handy to check. I... | 1 | https://mathoverflow.net/users/8133 | 337107 | 143,901 |
https://mathoverflow.net/questions/337020 | -4 | In the case of 1-categories, we know there is a functor category
$PSh(C):=[C^{op},Set]$, where $C$ is a small category,
and this functor category is a topos. I am hoping this will extend to the case of 2-categories and especially the 2-category of groupoids.
| https://mathoverflow.net/users/10007 | Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids? | Gpd is the 2-category of groupoidd.
The category of 2-functors and natural transformations $Gpd^{op} \rightarrow C$, for $C$ any 1-category is equivalent to the category $Gpd'^{op} \rightarrow C$ of functors and natural transformations, where $Gpd'$ has the same objects as $Gpd$ but isomorphism classes of functors as... | 1 | https://mathoverflow.net/users/10007 | 337108 | 143,902 |
https://mathoverflow.net/questions/336999 | 6 | If I have a linked pair of circles in $\mathbb{R}^3$, they can be unlinked in $\mathbb{R}^4$. Said differently, there is an isotopy in $\mathbb{R}^4$ between two strands which have been twisted, and two untwisted strands. Here is an ascii art picture!
```
a b a b
| | | |
\ / | ... | https://mathoverflow.net/users/8639 | unlinking in 5 dimensions | Thanks, I understand your question now. I asked a similar question in an old paper of mine, but it was with knots rather than braids. In short, the answer is **yes** they are isotopic in $\mathbb R^4$ subject to certain carefulness assumptions. But if you are particularly fussy about the set-up there is a way in which ... | 6 | https://mathoverflow.net/users/1465 | 337111 | 143,904 |
https://mathoverflow.net/questions/337047 | 11 | Say that $S\subset \mathbb Z\_n$ is *stabbed* by $X\subset \mathbb Z\_n$ if for every $t$ we have $(S+t)\cap X\ne \emptyset$.
>
> Is there for every $|S|=4$ an $|X|<n/2$ that stabs it?
>
>
>
My motivation comes from [here](https://dustingmixon.wordpress.com/2019/07/08/polymath16-thirteenth-thread-bumping-the-d... | https://mathoverflow.net/users/955 | Is it possible to stab (every rotation of) any four element subset of $\mathbb Z_n$ with less than $n/2$ elements? | I think this is true for any group $G$, and if $|G|$ is odd then $|S| \geq 3$ suffices. I will use multiplicative notation instead of additive notation since I do not assume that $G$ is commutative.
**Theorem.** If $G$ is a finite group and $S \subseteq G$ with $|S| \geq 4$ then there is a set $X \subseteq G$ such th... | 10 | https://mathoverflow.net/users/2000 | 337113 | 143,905 |
https://mathoverflow.net/questions/337065 | 1 | 1) How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain (Volume)?
2) How many nodes $v$ does a ball with center $x$ of radius $r$ in the Johnson graph $J(n,k)$ contain such that $d(x,v)=r$ (Surface)?
The Jonhson Graph is defined here:
<https://en.wikipedia.org/wiki/Johnson_graph>
Any r... | https://mathoverflow.net/users/nan | How many nodes does a ball of radius $r$ in the Johnson graph $J(n,k)$ contain? | The number of vertices $p\_r$ which satisfy $d(x,v)=r$ in a distance regular graph can be computed from the intersection array:
$p\_r=\frac{b\_1b\_2...b\_r}{c\_1c\_2...c\_r}$
For Johnson graphs, $p\_r=(^k\_r)(^{n-k}\_{ r})$.
| 4 | https://mathoverflow.net/users/125498 | 337125 | 143,911 |
https://mathoverflow.net/questions/17819 | 15 | $\mathfrak{sl}\_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = -2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for "Cartan.") My guess, based on similarities between these commutator relations and ones I have seen mentioned when people ... | https://mathoverflow.net/users/290 | What's the origin of the naming convention for the standard basis of $\mathfrak{sl}_2(\mathbb{C}) $? | The letters $\mathrm X$ and $\mathrm Y$ are already used by **Cayley** in what Dieudonné (in [MR](//ams.org/mathscinet-getitem?mr=88c:01020)) calls the first description of all finite-dimensional irreducible $\mathfrak{sl}\_2$-modules: *A Second Memoir upon Quantics* ([1856](//doi.org/10.1098/rstl.1856.0008), §§29–31).... | 16 | https://mathoverflow.net/users/19276 | 337130 | 143,915 |
https://mathoverflow.net/questions/337138 | 7 | Let $(M,\xi)$ be a transversally orientable contact manifold, that is, there exists a form $\alpha \in \Omega^1(M)$ such that $\xi = \ker \alpha$. Then we can associate to $(M,\xi)$ its **symplectisation** $(\mathbb{R} \times M,d(e^t\alpha))$, a symplectic manifold. I wondered, if there is a categorical setting for thi... | https://mathoverflow.net/users/98139 | Symplectisation as a functor between appropriate categories | first of all I think your $S(F)$ can be modified into
\begin{align\*}
S(F)(t,x)=(t-\log(|f(x)|), F(x))
\end{align\*}
since $f$ is non-vanishing, this is always smooth. Nevertheless, there is a more conceptual way to see the symplectization:
the symplectization $S$ is a functor from contact manifolds into homogeneous s... | 7 | https://mathoverflow.net/users/135670 | 337141 | 143,918 |
https://mathoverflow.net/questions/337140 | 3 | I raise this confusing because I try to understand the witt vectors for characteristic not equal to p.
Let us assume p=2. The Witt Polynomials is explicitly given by
$$
S\_0=X\_0+Y\_0
$$
$$
S\_1=X\_1+Y\_1-X\_0Y\_0
$$
Then consider truncated Witt vectors $W\_2(\mathbb{Z}[X])$ for example. Let's add two vectors
... | https://mathoverflow.net/users/98747 | Witt vectors addition confusing | There is nothing wrong, the point of confusion as you noted lies in the characteristic of the ground ring $R = \mathbf{Z}[X]$. When $R$ has characteristic $0$, then the ghost map sending $W(R) \to R^\mathbf{N}$ is invertible whence an isomorphism. If you want or need more details on the ghost map, recall that $W\_n(\un... | 4 | https://mathoverflow.net/users/13288 | 337151 | 143,922 |
https://mathoverflow.net/questions/315165 | 1 | Let $G = (V(G), E(G))$ be an infinite connected simple graph.
Let $((S\_n)\_n, (P^x)\_{x \in V(G)})$ be the simple random walk on $G$.
Let $p\_n (x,y) = P^x (S\_n = y)$.
A spectral dimension of $G$ is given by the following:
$$ d(G) = -2\lim\_{n \to \infty} \frac{\log p\_{2n}(x,x)}{\log n}, x \in V(G)$$
if the limit... | https://mathoverflow.net/users/116429 | Is there a transient graph whose spectral dimension two? | Consider the wedge $\{(x,y,z) \in {\bf Z}^3 : |z| \le \log^2(|x|+2)\}$. It is transient by
[1] yet close enough to the two dimensional lattice so that it has spectral dimension 2.
[1] T.J Lyons : A Simple Criterion for Transience of a Reversible Markov Chain
<https://projecteuclid.org/euclid.aop/1176993604>
| 1 | https://mathoverflow.net/users/7691 | 337159 | 143,927 |
https://mathoverflow.net/questions/337134 | 1 | Let $\sigma>0$ and $\mathcal N\_{x,\:\sigma^2}$ denote the normal distribution with mean $x\in\mathbb R$ and variance $\sigma^2$. From the [Ionescu-Tulcea theorem](https://en.wikipedia.org/wiki/Ionescu-Tulcea_theorem), we know that $$\kappa(x,\;\cdot\;):=\bigotimes\_{n\in\mathbb N}\mathcal N\_{x\_n,\:\sigma^2}$$ is a w... | https://mathoverflow.net/users/91890 | Can we show that $\mathbb R^{\mathbb N}\ni x\mapsto\bigotimes_{n\in\mathbb N}\mathcal N_{x,\:\sigma^2}$ is a Markov kernel? | First of all, I suppose you mean $\kappa$ to be defined as
$$\kappa(x,\;\cdot\;):=\bigotimes\_{n\in\mathbb N}\mathcal N\_{x\_n,\:\sigma^2}$$
with $x\_n$ on the right side instead of $x$, where $x = (x\_1, x\_2, \dots)$. As originally written it didn't make sense.
Defined thus, $\kappa$ is indeed a Markov kernel. As y... | 1 | https://mathoverflow.net/users/4832 | 337162 | 143,928 |
https://mathoverflow.net/questions/337164 | 5 | Let $p:S^3 \to S^2$ be the Hopf fibration which is a result of the standard action of $S^1$ on $S^3$.
>
> Is there a $2$ dimensional vector bundle $\tilde{p}:E \to S^2$ such that $S^3\subset E$ and $\tilde{p}\_{|S^3}=p$?
>
>
> Is there a vector bundle $E$ as above with the following extra condition:The total spac... | https://mathoverflow.net/users/36688 | Extension of Hopf fiber bundle to (an equivariant) 2 dimensional vector bundle | I think both of your questions can by answered positive. Since the Hopf fibration is an $S^1$ bundle it comes as a sphere bundle of complex line bundle over $S^2$. From the Gysin sequence the first Chern class of that line bundle must be a generator of $H^2(S^2;\mathbb Z)$ which determines the isomorphism type of the l... | 7 | https://mathoverflow.net/users/20999 | 337169 | 143,931 |
https://mathoverflow.net/questions/337168 | 7 | Let $T:L^2(\mu)\to L^2(\mu)$ be a linear and continuous operator, where $L^2(\mu)$ is the (real) $L^2$-space to some $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$.
$T$ is assumed to be sign-preserving in the sense that
$$
v(x) \cdot (Tv)(x) \ge0
$$
for $\mu$-almost all $x\in \Omega$ and all $v\in L^2(\mu)$.
... | https://mathoverflow.net/users/48485 | Is a sign-preserving operator on $L^2$ a multiplication? | First, if $uv=0$ then $uTv=0$ a.e.: indeed, applying the hypothesis with the function $\epsilon u+\epsilon^{-1}v$ and evaluating on $\{u\neq 0\}$, we get
$$0\le \epsilon^2uTu+\epsilon^{-2}vTv+uTv+vTu=\epsilon^2 uTu+uTv$$
a.e. and deduce the claim (sending $\epsilon\to 0$). This gives the property $(\*)$ that you showed... | 5 | https://mathoverflow.net/users/36952 | 337174 | 143,932 |
https://mathoverflow.net/questions/337118 | 0 | Consider $X = (\mathbb{R^n},c)$, where $c$ is the equivalence class of all torsion free affine connections having straight lines as unparameterized geodesics. What is the group of symmetries of $X$? This ought to be known. Thank you.
| https://mathoverflow.net/users/81645 | What is the group of symmetries of $\mathbb{R^n}$ with the flat projective structure? | Long answer taking a complicated but rewarding detour: This is example that fits into the framework of Cartan geometries. Category of manifolds with an equivalence class of torsion free affinne connections that have the same unparametrized geodesics is equivalent to (properly normalized) Cartan geometry modeled on a ho... | 1 | https://mathoverflow.net/users/6818 | 337181 | 143,934 |
https://mathoverflow.net/questions/337144 | 7 | In papers published 1920 and 1922, Skolem offered two separate proofs of a result due to Lowenheim. On this basis we can distinguish a strong and a weak version of the Lowenheim-Skolem theorem as follows:
The weak (1922) version states that if a closed formula $\phi$ of quantification theory is satisfiable, then it ... | https://mathoverflow.net/users/116705 | Equivalence between Lowenheim-Skolem Theorem and Godel Completeness | On Question (a), yes this is correct. On Question (b), almost yes. If you drop the assumption that ϕ is satisfiable, then Skolem has given a sound and complete procedure for refuting a formula in a finite number of steps. However it is not (and cannot be) also effective. If the formula is in fact not refutable then the... | 6 | https://mathoverflow.net/users/38783 | 337185 | 143,935 |
https://mathoverflow.net/questions/337188 | 6 | Consider a [primitive permutation group](https://en.wikipedia.org/wiki/Primitive_permutation_group) $\Gamma\subseteq\mathrm{Sym}(N)$ on the $n$-element set $N=\{1,...,n\}$, that is, $\Gamma$ does not preserve any non-trivial partition of $N$.
Consider the linear representation of $\Gamma$ by permutation matrices on $... | https://mathoverflow.net/users/108884 | Irreducible factors of primitive permutation group representation | An example in which there are two isomorphic irreducible modules in the decomposition is the group ${\rm PSL}(2,11)$ in its primitive permutation representation of degree $55$ coming from the action of $G$ on the cosets of a dihedral subgroup of order $12$. The permutation module over the real numbers decomposes into m... | 4 | https://mathoverflow.net/users/35840 | 337190 | 143,937 |
https://mathoverflow.net/questions/337189 | 1 | Suppose that $G$ is a connected Lie group of unitary matrices and $U(t)\in G$ depends continuously differentiable on a real parameter $t$ and has no real eigenvalue -1. Then the principal value of the logarithm $W(t):=\log U(t)$ is also continuously differentiable in $t$.
If $G$ is abelian, we may conclude (with dot... | https://mathoverflow.net/users/56920 | A matrix derivative | The comment by Vít Tuček leads to the formula
$$\frac{d}{dt}\log(1+A(t))=\int\_0^1 (1+sA(t))^{-1}\dot A(t)(1+sA(t))^{-1} ds.$$
This can be verified by Taylor expanding both sides if the norm of $A(t)$ is less than 1, and follows in general by analytic continuation when no real eigenvalue of $A(t)$ is $\le -1$. (The pro... | 1 | https://mathoverflow.net/users/56920 | 337192 | 143,939 |
https://mathoverflow.net/questions/337201 | 1 | Let $\mathcal{C}=\{M\_i\}$ be a Fraïssé class of finite $\mathcal{L}$-structures with the generic model $M$. Also, let $M^\*=\prod\_U M\_i$ the be ultraproduct of members of $\mathcal{C}$ where $U$ is a non-principal ultrafilter.
Question. What is the relation between $M$ and $M^\*$?
| https://mathoverflow.net/users/nan | Fraïssé Limit and Ultraproduct | First note that $M$ is countable while $M^\*$ has size continuum. So the two structures are never isomorphic. But you may instead ask when/if they are elementarily equivalent; and the answer is “sometimes”.
Example 1: Let $\mathcal{C}$ be the class of finite graphs. Then $M$ is the countable random graph, and one ca... | 4 | https://mathoverflow.net/users/38253 | 337204 | 143,943 |
https://mathoverflow.net/questions/337178 | 4 | Two days ago it occurred to me that almost all simple graphs are small world networks in the sense that if $G\_N$ is a simple graph with $N$ nodes sampled from the Erdös-Rényi random graph distribution with probability half then when $N$ is large:
\begin{equation}
\mathbb{E}[d(v\_i,v\_j)] \leq \log\_2 N \tag{1}
\end... | https://mathoverflow.net/users/56328 | Almost all simple graphs are small world networks |
>
> whether there is an elementary proof that almost all simple graphs are very small world networks
>
>
>
Following up on Brendan McKay's comment. The chance that an Edos-Renyi$(0.5,n)$ graph has diameter one is $0.5^{n(n-1)/2}$, which of course goes to zero exponentially fast.
On the other hand, two vertices... | 1 | https://mathoverflow.net/users/29697 | 337206 | 143,944 |
https://mathoverflow.net/questions/337216 | 3 | As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root.
Are there classical equations of the form
$$\sum\_{i=0}^{\infty} a\_ix^i =1$$
that have $e$ or $1/e$ as root, with $a\_i\in \mathbb{Z}$ for each $i$?
For $1/e$, is it possible to have $a\_i\ge 0$ for each $... | https://mathoverflow.net/users/135096 | Power series equation with solution $1/e$ | If $a\_i=0$ for all large enough $i$, then $e$, being transcendental, cannot be a solution to your equation, unless $a\_i=0$ for all $i\ge1$. Otherwise, if $a\_i\ne0$ for infinitely many $i$, then the series for $e$ cannot converge, given that the $a\_i$'s are integers, because then $|a\_ie^i|\ge e^i\not\to0$ if $a\_i\... | 5 | https://mathoverflow.net/users/36721 | 337218 | 143,949 |
https://mathoverflow.net/questions/337183 | 6 | Given a right Hilbert $A$-module $E$, and a right Hilbert $B$-module $F$, together with non-degenerate $\*$-homomorphism $\phi:A \to \mathcal{L}\_B(F)$, we can form the tensor product
$$
E \otimes\_{\phi} F,
$$
by completing the $A$-balanced tensor product of $E$ and $F$ and completing with respect to the obvious norm.... | https://mathoverflow.net/users/128876 | Tensoring adjointable maps on Hilbert modules | That $K$ is a bimodule map means that $\phi(a)K(x) = K(\phi(a)(x))$ for all $a\in A, x\in F$. That is, $K \in \phi(A)' \subseteq \mathcal{L}\_B(F)$.
I am here following the ideas of Lance's little Hilbert $C^\*$-modules book, Chapter 4. Let $z = \sum\_i x\_i\otimes y\_i \in E\odot F$, so $(z|z) = \sum\_{i,j} (y\_i|\p... | 4 | https://mathoverflow.net/users/406 | 337222 | 143,951 |
https://mathoverflow.net/questions/337100 | 10 | Let $X$ be a real $n$ dimensional manifold. One knows that it can be embedded into $\mathbb{R}^{2n}$ by the Whitney embedding theorem. The normal bundle for such an embedding will be a rank $n$ real vector bundle. I would be interested in understanding the connection between this embedding, its normal bundle and the ra... | https://mathoverflow.net/users/109370 | Normal bundle of Whitney embedding | This is to shed some light on Part 3 which asks for a classification of normal vector bundles of a smooth $n$-dimensional submanifold $X$ of $\mathbb R^{2n}$.
The normal bundle $\nu$ to $X$ is stably isomorphic to the negative of $TX$, and in particular the Pontryagin and Stiefel-Whitney classes of $\nu$ are complet... | 5 | https://mathoverflow.net/users/1573 | 337225 | 143,952 |
https://mathoverflow.net/questions/337205 | 0 | For example, is there a function $g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ such that the condition $\int\_\mathbb{R}\int\_\mathbb{R}|f(u,v)g(u,v)|dudv<\infty$ implies $\sum\limits\_{k\in\mathbb{Z}}\int\_\mathbb{R}\int\_\mathbb{R}e^{ik(u+v)}f(u,v)dudv<\infty$?
| https://mathoverflow.net/users/61204 | For which $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, $\sum\limits_{k\in\mathbb{Z}}\int_\mathbb{R}\int_\mathbb{R}e^{ik(u+v)}f(u,v)dudv<\infty$? | Writing $g(t) = \int\_{-\infty}^\infty f(t+s,-s) ds$, we reduce the question to the following one: when $$\sum\_{k = -\infty}^\infty \int\_{-\infty}^\infty e^{i k t} g(t) dt$$ is finite? Setting $h(t) = \sum\_{k = -\infty}^\infty g(t + 2 \pi k)$, we see that the summands in the above expression are simply the coefficie... | 3 | https://mathoverflow.net/users/108637 | 337242 | 143,955 |
https://mathoverflow.net/questions/221048 | 5 | I am interested in evaluating some elliptic integrals, and I have not been able to secure a reference to do exactly what I need. Most of the references I've found seem to focus on reducing more general elliptic integrals to Legendre form, but leave out the part about actually dealing with complete elliptic integrals. I... | https://mathoverflow.net/users/10898 | Evaluating elliptic integrals | This question has been out for a while, and I think it deserves a thorough answer, even tho I came quite late to this party.
As both the classical Legendre-Jacobi theory and the Carlson theory have been mentioned by other users, I'll treat the OP's integral from both viewpoints.
---
Legendre-Jacobi
------------... | 11 | https://mathoverflow.net/users/7934 | 337251 | 143,958 |
https://mathoverflow.net/questions/337078 | 1 | Closely related (although not equivalent) to minimax optimization problems is the following:
$$\min\_{x \in \Omega} \min\_{i=1,...,q} f\_i (x).$$ Here, $\Omega \subset \Bbb R^n$ and $f\_i: \Bbb R^n \to \Bbb R$ is continuously differentiable. I am looking for references on algorithms for this kind of problems. Specifi... | https://mathoverflow.net/users/114128 | On solution methods for min-min optimization problems | This comment is too long so I directly post it as an answer. It is not hard to propose such a method.
**Assumption**: Every $f\_i(x)$ is Lipschitz differentiable with constant $L$, and define $h(x):=\min\_{i=1,...,q}f\_i(x)$. Suppose further that $h(x)$ is bounded from below.
First, it is easy to prove:
**Optim... | 1 | https://mathoverflow.net/users/113353 | 337252 | 143,959 |
https://mathoverflow.net/questions/337214 | 9 | A group G is said to have a *factorization* if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \ | \ a \in A, b \in B \}$.
The paper [Factorisations of sporadic simple groups](https://doi.org/10.1016/j.jalgebra.2006.04.019) (by Michael Giudici) provides the classification of all the factorizations... | https://mathoverflow.net/users/34538 | Groups without factorization | This must be "well-known": If we have $G = AB$ when $G$ is a finite group, and $A,B$ are proper subgroups of $G$, then we may suppose that $A$ and $B$ are both maximal.
For if $A$ is not maximal, and $A < C$ with $C$ maximal, then we still have $G = CB,$ and $B \not \leq C$, so we may replace $A$ by $C$ and assume th... | 17 | https://mathoverflow.net/users/14450 | 337256 | 143,961 |
https://mathoverflow.net/questions/337255 | 3 | Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ we have for every $\eta \geq \omega$ that $\eta \in A$. Now a percolation measure $\mu$ satisfies the FKG inequality if ... | https://mathoverflow.net/users/143779 | Does the union of two percolation measures satisfying the (FKG) inequality still satisfy (FKG)? | Welcome to MathOverflow! The answer to your question is yes. Indeed, let $X$ and $Y$ be independent random elements of the set $\{0,1\}^n$ whose distributions are percolation measures, which latter thus satisfy the [Harris--FKG inequality, Proposition 2.3](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2... | 2 | https://mathoverflow.net/users/36721 | 337258 | 143,962 |
https://mathoverflow.net/questions/336969 | 5 | Do we know of any closed symplectic manifold $M$ with 2 cohomologous symplectic forms $\omega\_1$ and $\omega\_2$ such that there exist $\psi \in \text{Diff}(M)$ and $\psi^\* \omega\_1 = \omega\_2$ but $\omega\_1$ and $\omega\_2$ are not isotopic?
Here the word isotopy is supposed to mean that the two forms a joined ... | https://mathoverflow.net/users/92483 | Diffeomorphic but not isotopic symplectic forms | This answer is an extension of my last comment, which in turn is just a reference to [this answer](https://mathoverflow.net/a/34249/35687) by MO user Petya, and the paper [Symplectic Topology and Capacities](http://www.math.stonybrook.edu/~dusa/princerev98.pdf) by McDuff cited therein.
First, your condition for two f... | 3 | https://mathoverflow.net/users/35687 | 337260 | 143,963 |
https://mathoverflow.net/questions/337262 | 1 | Let $\alpha<0$ and let $u(x):= e^{\alpha x}$ for $x \geq 0$. I'm reading a paper which states that there are constants $d\_{j}, \beta\_{j} \in \mathbb{C}$ with $\beta\_j>0$ such that if we define
$$v(x) =\sum\_{j=1}^n d\_{j} e^{ \beta\_{j} (x-b)}$$
then $w^{(j)}(b)=0$ for $j=0,1, \ldots, n-1$ if $w(x):=v(x)+u(x)$, and
... | https://mathoverflow.net/users/143016 | Show that $\| v\|_{L^2(0,b)}=o(\| u\|_{L^2(0,b)})$ as $b \to \infty$ | Not sure if I understood correctly the question, but this is too long for a comment.
Fix $\alpha < 0$ and take any distinct $\beta\_j > 0$, $j = 1, 2, \ldots n$. Then there is a unique solution $d\_j$, $j = 1, 2, \ldots, n$, of the system of linear equations $$ \frac{d^i}{dx^i} \bigg|\_{x = 0} \biggl(\sum\_{j = 1}^n ... | 1 | https://mathoverflow.net/users/108637 | 337267 | 143,964 |
https://mathoverflow.net/questions/336863 | 1 | Let $\mu\_1$ and $\mu\_2$ be 1D gaussian distributions with means $m\_1$ and $m\_2$ respectively and common variance $\sigma$. Let $\Omega$ be a closed subset of $\mathbb R^2$, and consider the cost function $c\_\Omega:\mathbb R \times \mathbb R \rightarrow \{0, 1\}$ defined by $c\_\Omega(x',x) = 1$ if $(x', x) \in \Om... | https://mathoverflow.net/users/78539 | Simplify Wasserstein distance between Gaussians with binary cost function | This is an answer to the second question, for the particular choice of $\Omega$. In fact, in order to make the notation slightly simpler, let us assume that $\Omega = \{(x\_1, x\_2) : |x\_1 - x\_2| > \alpha\}$, with a strict inequality. By a simple approximation argument one easily sees that this change does not influe... | 3 | https://mathoverflow.net/users/108637 | 337285 | 143,969 |
https://mathoverflow.net/questions/337276 | 1 | Let $A\in\mathbb{R}^{n^k}$ be a $k$-dimensional tensor with $n$ elements along each dimension. Moreover suppose $u\_1,u\_2,\dots,u\_k\sim\text{Unif}(\pm1)^n$ are $n$ dimensional vectors with each of their components drawn iid from Rademacher distribution, and let $U:=u\_1\otimes u\_2 \otimes \dots \otimes u\_k$ be a ra... | https://mathoverflow.net/users/11363 | Sketching Frobenius norm of a tensor with a rank-1 random tensor | The tail probability $P(X\ge x)$ is in general like $e^{-x^{2/k}}$ for large $x>0$.
Indeed, we have
\begin{equation\*}
X=\sum\_{i\_1,\dots,i\_k\le n} a\_{i\_1,\dots,i\_k}u\_{1,i\_1}\dots u\_{k,i\_k},
\end{equation\*}
where the $u\_{i,j}$'s are assumed to be independent Rademacher random variables.
Without loss o... | 1 | https://mathoverflow.net/users/36721 | 337287 | 143,971 |
https://mathoverflow.net/questions/337187 | 2 | If $\prod\_{i=1}^t x\_i^{e\_i}$ is a monomial, define
$$rad\biggl(\prod\_{i=1}^t x\_i^{e\_i}\biggr)$$
to be the number of distinct (nonzero) values of $e\_i$.
Now let $G$ be a simple graph with vertices labeled by integers, and consider the graph polynomial
$$P\_G := \prod\_{i<j}(x\_i-x\_j)$$
where the product is over ... | https://mathoverflow.net/users/100231 | Chromatic number and graph polynomial | $G=K\_{3,3}$ is a counterexample: it has chromatic number $2$ but $\mathrm{rad}(P\_G)=3$; there are monomials with all three exponents $1,2,3$.
My conjecture would be that $\mathrm{rad}(P\_G)$ is equal to the (maximum) degree of $G$ if $G$ is regular.
---
**Edit:**
I claim that if $G$ is a bipartite $k$-regu... | 4 | https://mathoverflow.net/users/24076 | 337289 | 143,972 |
https://mathoverflow.net/questions/337277 | 12 | **Background and notations:**
Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a *Thom space* $T\_n(\xi)$, given by the cofiber:
$\require{AMScd}$
\begin{CD}
\ \xi @>>> B\\
@VVV @VVV\\
\* @>>> T\_n(\xi).
\end{CD}
This const... | https://mathoverflow.net/users/125244 | Equivalent definitions of Thom spectra | The details of the comparison are treated in detail in the original ABGHR paper (and then unfortunately split in half across two papers in the updated version), so I'll just try to give a sketch of what's going on.
Thom spaces
-----------
Given any type of bundle $E \to B$ we can view this as a diagram of spaces in... | 14 | https://mathoverflow.net/users/6936 | 337299 | 143,975 |
https://mathoverflow.net/questions/336833 | 1 | This is a [cross-post](https://math.stackexchange.com/q/3299314/64809) from math.stackexchange.com. There has been no response there.
Given a Markov chain of finite states with constant transition probabilities, what is the method to compute the probability of a path finally landing in a particular state for the firs... | https://mathoverflow.net/users/32660 | Probability of traversing all other states and finally landing on one state | For the example, the considered event $E$ is the intersection of two events $A$ and $B$, namely that both 5 and 7 are reached before 6. The union of the two events has probability $1$. So the probability of $E$ is $P(A)+P(B)-1$. By this answer:
<https://math.stackexchange.com/questions/725996/reaching-a-level-before-... | 4 | https://mathoverflow.net/users/112954 | 337305 | 143,978 |
https://mathoverflow.net/questions/337310 | 1 | The following question seems intuitively true, but I'm unable to see the proof. While I could prove it when $U=\mathbb{R}^n$, but for other open sets, I do not have a proof. Although I could construct $\alpha:[0,1]\to\mathbb{R}^n$ satisfying 1, 2 and 3, I can't force it to remain in $\bar{U}$. Could you please let me k... | https://mathoverflow.net/users/27832 | Existence of Smooth path in a Domain through a Sequence of Points | In my other answer, I show how the result for $\mathbb{R}^N$ can be extended to arbitrary $U$. Here I prove that the result is actually *false* for $U = \mathbb{R}^N$ (so in a sense my other answer is completely void).
Consider $p = 0$ and $p\_n$ such that the shortest path that includes all $p\_n$ contained in the s... | 3 | https://mathoverflow.net/users/108637 | 337316 | 143,985 |
https://mathoverflow.net/questions/337322 | 3 | Let $G=(V,E)$ be a finite simple, undirected graph. Given $f:V\to \mathbb{Z}$ we define the *neighborhood
sum function* $\mathrm{nsum}\_f:V\to\mathbb{Z}$ by setting
>
> $\mathrm{nsum}\_f(v) = \sum\{f(w):w\in N(v)\}$ for all $v\in V$.
>
>
>
We say that a graph $G=(V,E)$ is *sum-balanceable* if there is an inje... | https://mathoverflow.net/users/8628 | Sum-balanceable finite graphs | Consider the graph with vertices $\{x,y\}\times\{ 1,2,...n\}$, and every two vertices connected unless they are of the form $(x,a),(y,a)$ $(a\in\{1,2,...n\})$.
The graph have chromatic number $n$, since the graph is perfect and its clique number is $n$.
Noticing that the vertex $(x,a)$ is connected to $(x,k)$ and $... | 3 | https://mathoverflow.net/users/125498 | 337340 | 143,993 |
https://mathoverflow.net/questions/337327 | 3 | Let $(a,b)$ be a pair of coprime positive integers with $a$ being even. Are these conditions sufficient to prove that there exist infinitely many positive integers $n,$ such that $(a^n+1,b^n+1)=1$ ?
| https://mathoverflow.net/users/104326 | Greatest common divisor of $(a^n+1,b^n+1)$ | I think that this sort of question was originally asked by Ailon and Rudnick, but they use $-1$ instead of $+1$ and asked if $\gcd(2^n-1,3^n-1)=1$ for infinitely many $n$. In this setting, for more general $a$ and $b$, the right question/conjecture would be $\gcd(a^n-1,b^n-1)=\gcd(a-1,b-1)$. They prove something strong... | 12 | https://mathoverflow.net/users/11926 | 337342 | 143,994 |
https://mathoverflow.net/questions/337278 | 9 | In "The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)" by Haim Judah and Saharon Shelah
JSL Vol. 55, No. 3 (Sep., 1990), pp. 909-927 ([JdSh308] in Shelah's numbering) the authors remark on the final page that adding a pair of Hechler reals makes the union o... | https://mathoverflow.net/users/114946 | Effect of adding one Hechler real versus adding two on the meager ideal | If $c$ is Cohen over $V$, and $d$ is dominating over $V[c]$ (not necessarily Hechler-generic), then in $V[c][d]$ there is a meager set covering all meager sets from $V$. Hence 2 successive Hechler reals make the union of all old meager sets meager.
(I suspect that this is not true if you just add one Hechler real.)
... | 9 | https://mathoverflow.net/users/14915 | 337343 | 143,995 |
https://mathoverflow.net/questions/337353 | 4 | For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}\_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, the theory of Schur functors.
What happens for the exceptional Lie algebras? For example, taking $V$, the fundamental r... | https://mathoverflow.net/users/125941 | Decomposing tensor powers of the fundamental representation of exceptional Lie algebras | If you're e.g. looking for an analog of Schur-Weyl duality in other types, the combinatorics can get very tricky very quickly. For the symplectic group I believe the answer was first worked out by Sundaram in her PhD thesis (<https://dspace.mit.edu/handle/1721.1/15060>) and meanwhile for the odd orthogonal group the an... | 2 | https://mathoverflow.net/users/25028 | 337354 | 143,998 |
https://mathoverflow.net/questions/336947 | 1 | If $G$ is a $\mathbb Q$-defined subgroup of $\operatorname{GL}\_n(\mathbb C)$, $\Lambda$ is a subgroup of $G(\mathbb Z)$, and $U$ is a unipotent subgroup of $G(\mathbb C)$ such that $\Lambda \cap U$ is Zariski-dense in $U$, why must $\Lambda$ contain $U(k\mathbb Z)$ (the principal congruence subgroup of $U(\mathbb Z)$ ... | https://mathoverflow.net/users/142244 | If $\Lambda \cap U$ is Zariski-dense in $U$, then $\Lambda$ contains $U(k\mathbb Z)$ for some $k ≥ 1$? | In this answer i will follow Yves' comment and add references. If $U = \mathbf{U}(R)$ with $\mathbf U$ an algebraic unipotent $\mathbb Q$-group then the two following facts hold :
* If $\Lambda \le U$ is Zariski-dense subgroup then $U/\Lambda$ is compact (Theorem 2.1 in Raghunathan's book *Discrete subgroups of Lie ... | 2 | https://mathoverflow.net/users/32210 | 337361 | 144,001 |
https://mathoverflow.net/questions/271778 | 11 | It is known that the [Somos-$k$ sequences](http://mathworld.wolfram.com/SomosSequence.html) for $k\ge 8$ do not give integers. But the first terms of **Somos-8** sequence $s\_n=a\_n/b\_n$
$$1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815,\frac{420514}{7}, \frac{28670773}{91}$$
defined by $s\_1=s\_2=s\_3... | https://mathoverflow.net/users/5712 | Is Somos-8 $\mod 2$ periodic? | I confirm the observation of @მამუკაჯიბლაძე that $\nu\_2(s\_{103})=-1$. That is, $s\_n\bmod 2$ is not well-defined at first place, which invalidates the question.
Moreover, for $n\geq 133$, $\nu\_2(s\_n)$ seems to form a strictly decreasing function, i.e., $s\_n$ accumulates larger and larger powers of $2$ in the den... | 8 | https://mathoverflow.net/users/7076 | 337371 | 144,003 |
https://mathoverflow.net/questions/337362 | 3 | Let $N \geq 3$ and let $\Gamma=\Gamma\_1(N)$, so that the moduli problem for elliptic curves with $\Gamma$-structure is fine. Let $Y=Y(\Gamma)$ be the corresponding moduli space.
In this context, one is usually given the definition of meromorphic forms and modular forms as follows. One has a universal elliptic curve... | https://mathoverflow.net/users/143589 | Questions about modular forms and the role of monodromy | Monodromy does not appear in such a nice way because $\omega$ is a coherent sheaf, not a locally constant sheaf. Thus there is not necessarily a way of continuing a local section along a path in $\mathbb H$.
One way to connect this to monodromy is to use the Eichler-Shimura isomorphism, which gives an isomorphism bet... | 4 | https://mathoverflow.net/users/18060 | 337372 | 144,004 |
https://mathoverflow.net/questions/337370 | 2 | All the definitions that follow is taken from [The Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf).
>
> **Definition 1.** Let $\bf{X}$ be a category. A *concrete category over $\bf{X}$* is a pair $({\bf{A}},U)$, where $\bf{A}$ is a category and $U :{\bf{A}} \to X$ is a faithful functor.
>
>
> **Defin... | https://mathoverflow.net/users/nan | How should I think about concrete functors and in particular about concrete isomorphism? | A concrete category should be thought of more or less as a category of structured sets (sets equipped with some sort of specified structure) and morphisms between them; the faithfulness to $Set$ means that morphisms are completely determined by their underlying functions.
For concrete isomorphism, let's take an exam... | 4 | https://mathoverflow.net/users/2926 | 337378 | 144,005 |
https://mathoverflow.net/questions/337393 | 2 | Let $\Sigma\_g$ be the a closed orientable surface of genus $g$.
My somewhat naive question: what is known about simple finite factors of $\pi\_1(\Sigma\_g)?$ In particular, I know that the composition series of this group is well understood in terms of homological algebra and topology: does this series filter $\pi\... | https://mathoverflow.net/users/7108 | What finite simple groups appear as factors of surface fundamental groups? |
>
> It appears that you are asking which groups $G$ occur as quotients of the fundamental group $\Pi\_g=\pi\_1(\Sigma\_g)$ for some $g$. Then the answer is "all finitely generated $G$", where $g$ can be taken to be the rank of $G$, i.e. the least cardinality of a generating set. If $G$ is a finite simple group, then ... | 10 | https://mathoverflow.net/users/5740 | 337394 | 144,011 |
https://mathoverflow.net/questions/162020 | 59 | I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to this question, which I suspect is an open problem: Does a $2n \times 2n$ square have the most domino tilings among all region... | https://mathoverflow.net/users/1450 | Which region in the plane with a given area has the most domino tilings? | Some answers are given in P. A. and L. Jordan, [Enumeration of border strips and Weil-Peterson volumes](https://arxiv.org/abs/1805.09778). [*Journal of Integer Sequences*, Vol. **22** (2019), Article 19.4.5](https://cs.uwaterloo.ca/journals/JIS/VOL22/Alexandersson/alex4.html).
We look at a certain type of regions (co... | 9 | https://mathoverflow.net/users/1056 | 337401 | 144,013 |
https://mathoverflow.net/questions/337211 | 1 | I have a family of probability distributions on the $n$-dimensional sphere $\mathbb S^n \subset \mathbb R^{n+1}$ defined in the following way:
$D\_0$ is the uniform distribution, which is constructed by sampling $k$ points $z\_i \in \mathbb R^{n+1}$ from a normal distribution of mean $m\_0 = (0,0, \dots, 0)$ and cov... | https://mathoverflow.net/users/93775 | Entropy of probability distributions on a sphere | Distributions on the sphere is studied and used in *directional statistics*, see for instance the text [Directional Statistics](https://www.bookdepository.com/Directional-Statistics-Kanti-V-Mardia/9780471953333?ref=grid-view&qid=1564651041318&sr=1-1) by Kanti Mardia & Peter Jupp. The entropy of the uniform distribution... | 2 | https://mathoverflow.net/users/6494 | 337408 | 144,015 |
https://mathoverflow.net/questions/337421 | 2 | Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct Hamiltonian cycles on a convex polytope may be a useful proxy measure. Now, let's suppose that the asymptotic formula f... | https://mathoverflow.net/users/56328 | The number of Hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$ | For $N\ge 4$ and arbitrary $n\ge N+1$ there is a [polytope with edge graph $K\_n$](https://en.wikipedia.org/wiki/Neighborly_polytope), hence $f\_N(n)=n!$. So if $n\ge N+2$ then
$$\frac{f\_{N+1}(n)}{f\_N(n)} = 1.$$
Note also, that $f\_N(n)$ is not well-defined for $n\le N$. So talking about *arbitrarily large* $N$ f... | 3 | https://mathoverflow.net/users/108884 | 337427 | 144,023 |
https://mathoverflow.net/questions/337444 | 4 | It is well-known how to compute the density of square-free numbers, to get
$$ \lim\_{N\to\infty} \frac{\#\{ n \leq N : n \text{ square-free}\}}{N} = \frac{6}{\pi^2}.$$
What is the density of numbers such that both $n$ and $n+1$ are square-free?
In other words, what is
$$\lim\_{N\to\infty} \frac{\#\{ n \leq N : n(n+1)... | https://mathoverflow.net/users/81295 | Density of twin square-free numbers | See *NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS* by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, <https://doi.org/10.1093/qmath/os-18.1.178>
This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of suc... | 7 | https://mathoverflow.net/users/17773 | 337447 | 144,030 |
https://mathoverflow.net/questions/333273 | 8 | ### Background
Recall that the *$q$-analogue* $[n]\_q\in\mathbb Z[q]$ of a natural number $n\in\mathbb N$ is defined as
$$ [n]\_q := \frac{q^n -1}{q-1}$$
the idea being that formulas involving $q$ will specialize along $\mathbb Z[q]/(q-p^n)$ to counting formulas about finite fields, while specializing along $\mathbb ... | https://mathoverflow.net/users/18702 | Product of $q$-analogues | Recall Legendre's formula
$$ v\_p(n!) = \sum\_{s=1}^\infty\left\lfloor\frac n{p^s}\right\rfloor = \sum\_{r=0}^\infty a\_r[r]\_p $$
where $n = \sum a\_r p^r$ is the base-$p$ expansion of $n$.
A $q$-analogue of this formula is provided by Lemma 4.8 of [The $p$-completed cyclotomic trace in degree 2](https://arxiv.org... | 2 | https://mathoverflow.net/users/18702 | 337449 | 144,032 |
https://mathoverflow.net/questions/337300 | 7 | Let $\mathbb{P}$ be the set of all perfect (i.e., every node has incomparable successors) subtrees of the full binary tree $2^{<\omega}$. We can endow $\mathbb{P}$ with a Borel structure by considering it as a subspace of $2^{2^{<\omega}}$ with the product topology.
If $p\in\mathbb{P}$, let $[p]$ denote the set of in... | https://mathoverflow.net/users/16107 | Set of perfect subsets of a Borel set | I believe that the answer is **no**, the set $S\_B$ need not be Borel.
Edit: I've incorporated some of François' suggestions from the comments, which should clean up the proof.
Consider the following embedding of $\omega^{<\omega}$ into $2^{<\omega}$: For the first level, send the node $(0)$ to $(0,0)$, and for $n>... | 5 | https://mathoverflow.net/users/16107 | 337452 | 144,034 |
https://mathoverflow.net/questions/277513 | 7 | This is a very simple question coming from the observation that every (pre)sheaf category has the maximal Cisinski model structure on it. This is the Cisinski model structure with the smallest class of weak equivalences possible.
Now, it is natural to ask: what is the maximal Cisinski model structure on the most can... | https://mathoverflow.net/users/62782 | Maximal Cisinski model structure on simplicial sets | (Edit: after 2.5 years, I've finally typed up the details as a paper! <https://arxiv.org/abs/2201.13400> )
I think I've worked out a relatively nice description of the fibrant objects. I won't be able to include the full proofs here, but I will give the description and explain a bit about why it works. (I'm planning ... | 7 | https://mathoverflow.net/users/132451 | 337453 | 144,035 |
https://mathoverflow.net/questions/337265 | 15 | Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$?
Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\sigma>1/2$ and $|t|\leq 5/8$ (note: much more is known for $q\leq 200000$ or so), I can give a bound of the form
$$\left... | https://mathoverflow.net/users/398 | $|L'(1,\chi)/L(1,\chi)|$ | Suppose that $\chi(-1)=1$ and that all non-trivial zeros $\beta+i\gamma$of $L(s,\chi)$ with $|\gamma|\le 1/2$ are on the critical line $\beta=1/2$. Recall the Hadamard factorization formula (see Davenport Chapter 12) which gives
$$
\frac{L^{\prime}}{L}(s,\chi) = -\frac 12 \log \frac q\pi - \frac 12 \frac{\Gamma^{\pri... | 11 | https://mathoverflow.net/users/38624 | 337456 | 144,038 |
https://mathoverflow.net/questions/337464 | 4 | Let $\mathbb{N}$ denote the set of positive integers, and let $G=(V,E)$ be a finite simple, undirected graph. Given $f:V\to \mathbb{Z}$ we define the *neighborhood
sum function* $\mathrm{nsum}\_f:V\to\mathbb{Z}$ by setting
>
> $\mathrm{nsum}\_f(v) = \sum\{f(w):w\in N(v)\}$ for all $v\in V$.
>
>
>
We say that ... | https://mathoverflow.net/users/8628 | Is sum-balanceability computable? | Yes, there is such a computable function $b$, which follows from the fact that one can compute solutions to [integer programs](https://en.wikipedia.org/wiki/Integer_programming). Note that we can model sum-balanceability by first introducing an integer variable $x\_v$ for each vertex $v \in V(G)$ and writing down the $... | 5 | https://mathoverflow.net/users/2233 | 337470 | 144,040 |
https://mathoverflow.net/questions/337032 | 10 |
>
> Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
>
>
>
I was told some fact along this line is true but could not find any reference after searching for a while.
| https://mathoverflow.net/users/21929 | Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group? | (Comments converted into an answer:)
Mostow (to whom I think this is often attributed?) gives a detailed proof in ([1949](//ams.org/mathscinet-getitem?mr=32656), Lemmas 2.2, 2.3).
Borel ([2001](//ams.org/mathscinet-getitem?mr=2002g:01010), pp. 152, 114) notes that algebraicity of perfect (e.g. semisimple) linear Li... | 6 | https://mathoverflow.net/users/19276 | 337474 | 144,041 |
https://mathoverflow.net/questions/337477 | 8 | Suppose $n\_p(G)$ is the number of elements of order $p$ in a group $G$.
Does there for every prime $p$ exist a $\epsilon\_p > 0$, such that for any group $G$ $n\_p(G) > (1 - \epsilon\_p(G))|G|$ implies, that $G$ is a $p$-group?
This question was inspired by a well known fact, that for any group $G$ $n\_2(G) > \fr... | https://mathoverflow.net/users/110691 | A question on "$p$-group bounds"? | Yes, this is the main theorem of
*Laffey, Thomas J.*, [**The number of solutions of (x^p=1) in a finite group**](http://dx.doi.org/10.1017/S0305004100052865), Math. Proc. Camb. Philos. Soc. 80, 229-231 (1976). [ZBL0343.20006](https://zbmath.org/?q=an:0343.20006).
Namely, if a finite group $G$ has more than $\tfrac... | 18 | https://mathoverflow.net/users/297 | 337483 | 144,042 |
https://mathoverflow.net/questions/337034 | 3 | **Context**: I am trying to show the convergence of an optimization method which includes orthogonalization in the update step.
**Problem**: Let's say I have real matrices $A, B \in \mathcal{R}^{n xm}$. If important, $n \gg m$ and I also know that the rank of the matrices is $m$. I can also assume that the norm of e... | https://mathoverflow.net/users/38479 | Bounding the Frobenius norm of orthogonalised matrices | There is no such bound. Let $u,v,w$ be orthonormal vectors. The matrices $A=[u,u+\varepsilon v]$ and $B=[u,u+\varepsilon w]$ have Q-factors $Q\_A=[u,v]$ and $Q\_B = [u,w]$ respectively, so $\|Q\_A-Q\_B\|$ is constant, but $\|A-B\|$ can be made arbitrarily small. You need to involve the condition number of $A$ and $B$ s... | 3 | https://mathoverflow.net/users/1898 | 337486 | 144,044 |
https://mathoverflow.net/questions/337469 | 3 | A hyperplane of a cube complex $X$ is a connected component of taking an $(n-1)$-cube for each midcube of $X$ and identifiying midcubes along faces of adjacent $n$-cubes of $X$.
If a group $G$ acts on a finite dimensional CAT(0) cube complex not cocompactly (with perhaps extra conditions), must there be an infinite ... | https://mathoverflow.net/users/143896 | Non-cocompact action on CAT(0) cube complex and hyperplane stabilizers | Actually, having finitely many orbits of hyperplanes is quite common. For instance:
**Proposition:** (Sageev) Let $G$ be a finitely generated group acting on a CAT(0) cube complex $X$. Then there exists a $G$-invariant convex subcomplex $Y \subset X$ containing only finitely many $G$-orbits of hyperplanes.
*Sketch... | 3 | https://mathoverflow.net/users/122026 | 337489 | 144,046 |
https://mathoverflow.net/questions/337500 | 5 | I was interested in an integral that I known from [1], it is
$$\int\_0^1 \log(x!)dx.$$
I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram Alpha online calculator. But I don't know how get the sum of certain series involving a special function.
My first step... | https://mathoverflow.net/users/142929 | On the integral $\int_0^1\log(x!)dx$ revisited | Details of the simple [integration by series](https://mathoverflow.net/questions/32954/multiplicative-integral-of-gammax#comment75191_32954) for $\int\_0^1\log(x!)dx$ mentioned [above](https://mathoverflow.net/questions/337500/on-the-integral-int-01-logxdx-revisited#comment843736_337500) (hopefully yours may be treated... | 6 | https://mathoverflow.net/users/6101 | 337517 | 144,059 |
https://mathoverflow.net/questions/334208 | 10 | *Notation:* Let $\phi$ be any formula in $\mathsf{FOL}({=},{\in}, W)$; let $\varphi$ be any formula in $\mathsf{FOL}({=},{\in})$ having $x$ free, and whose parameters are among $x\_1,\dotsc,x\_n$.
Note: “$W$” is a primitive constant symbol.
$\DeclareMathOperator\elm{elm}$Define: $\elm(y)\iff \exists z (y \in z)$, w... | https://mathoverflow.net/users/95347 | What's the exact consistency strength of this axiom system for classes and sets? | In regards to your comments, if you have two infinite theories $T$ and $T'$, if $T'$ consists of $T$ plus the schema $M\vDash T$ for some constant $M$, without alternate assumptions (Such as the existence of a truth predicate) $T'\nvdash Con(T)$. This is why $ZFC+V\_\kappa\prec V\nvdash V\_\kappa\vDash ZFC$.
As for y... | 2 | https://mathoverflow.net/users/141402 | 337521 | 144,061 |
https://mathoverflow.net/questions/337492 | 4 | Let $X$ be a normal projective variety with klt singularities with numerically trivial canonical divisor $K\_X$.
Does there always exist a Kähler-Einstein metric on $X$?
| https://mathoverflow.net/users/nan | Kähler-Einstein metrics on singular varieties | Yes, this is a result of Essydieux-Guedj-Zeriahi: "Singular Kähler-Einstein metrics" J. Amer. Math. Soc. 22 (2009), 607-639.
| 5 | https://mathoverflow.net/users/22294 | 337525 | 144,062 |
https://mathoverflow.net/questions/337537 | 6 | Let $f:A \to B$ be a finite flat local homomorphism of noetherian local rings.
>
> Are there some nice conditions on $A$ and $B$ which guarantee that the dimension
> of the Zariski tangent space of $A$ (at its maximal ideal) is smaller or
> equal to the dimension of the Zariski tangent space of $B$ (at its max... | https://mathoverflow.net/users/519 | Flat maps and Zariski tangent spaces | **Lemma.** Let $A \to B$ be a flat local homomorphism of Noetherian local rings with the same residue field $k$ such that $A$ and $B$ are both (local) complete intersections. Then
$$
\dim(B) - \dim(A) + \dim\_k \mathfrak m\_A/\mathfrak m\_A^2 \leq \dim\_k \mathfrak m\_B/\mathfrak m\_B^2
$$
provided $B$ is essentially o... | 10 | https://mathoverflow.net/users/143944 | 337545 | 144,069 |
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