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https://mathoverflow.net/questions/430157 | 2 | This is a question related to
[Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?](https://mathoverflow.net/questions/429754/semistable-curves-of-genus-g-geq-2-form-an-artin-algebraic-stack-in-the-etale)
Let $\mathcal C\rightarrow \mathcal M^{ss}\_g$ be the universal curve ove... | https://mathoverflow.net/users/152042 | On the stack of semistable curves | I am posting an answer to correct my wrong comments above. The answer is based on the comment by user @johan.
For every $g\geq 0$,
denote by $\mathfrak{M}\_g$
the stack of genus-$g$ curves that are proper, geometrically connected, reduced, and at-worst-nodal. Denote the universal curve over this stack by the $1$-morp... | 3 | https://mathoverflow.net/users/13265 | 430215 | 174,279 |
https://mathoverflow.net/questions/429909 | 7 | The following problem is presented in the paper *Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye* :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ continuous] is Li-Yorke chaotic, does there exist a Cantor scrambled set?"
I would like to know the current status of th... | https://mathoverflow.net/users/490774 | State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?" | This question was answered recently by Geschke, Grebík, and Miller:
S. Geschke, J. Grebík, and B. D. Miller, "Scrambled Cantor sets," *Proceedings of the AMS* **149** ([link](https://arxiv.org/abs/2006.08277) to the arxiv version)
They show that if $X$ is analytic (i.e., the continuous image of a Polish space), and... | 3 | https://mathoverflow.net/users/70618 | 430218 | 174,280 |
https://mathoverflow.net/questions/429630 | 4 | I am looking for a reference or proof of a lemma (if it's true) or a counter-example otherwise. It goes as follows:
Let $B\_1$ and $B\_2$ are two concentric balls of radius $1$ and $2$ in some $n$-dimensional Euclidean space. Then for any $f$ and $F$, there exists a $C>0$ such that
$$
\text{if }\;\Delta f=\text{div}F... | https://mathoverflow.net/users/131004 | Reference or proof of a lemma in PDE | I would call this estimate "the classical Calderon-Zygmund estimate" but indeed it is hard to track down a statement for the right-side in divergence form. Usually it is stated as an estimate from $L^p$ to $W^{2,p}$, when what you want is $W^{-1,p}$ to $W^{1,p}$. This is unfortunate, because the latter is the more natu... | 5 | https://mathoverflow.net/users/5678 | 430219 | 174,281 |
https://mathoverflow.net/questions/430214 | 1 | Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int\_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int\_{\Bbb R}\int\_{\Bbb R}|\widehat{f}(\xi)f(x)|^2e^{2|x\xi|} d\xi dx<\infty.$$
Put $f(x)=P(x) e^{-t x^2}$ where $t>0$ and $P$ is a
polynomial and suppose $f$ verify th... | https://mathoverflow.net/users/172078 | Why we have $f=0$ | Fourier transform of $f(x)=e^{-tx^2}$ is $\hat{f}(\xi)=ce^{-\xi^2/(4t)}$.
Multiplication of $f$ by a polynomial results in applying
a differential operator with constant coefficients to $\hat{f}$, and for our $\hat{f}$ this is equivalent to multiplication on some polynomial. These polynomials play no
role in convergenc... | 3 | https://mathoverflow.net/users/25510 | 430223 | 174,282 |
https://mathoverflow.net/questions/380958 | 1 | How can I prove the following Liouville theorem without using the mean value property?
>
> If $u$ is harmonic on $\mathbb{R}^n$ and $\int\_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $u$ is constant.
>
>
>
The proof that I know indeed uses the mean value property for harmonic functions.
--... | https://mathoverflow.net/users/143757 | Prove Liouville theorem without using mean value property | Below we will prove, by purely energy methods, the following sharper statement: Let $u$ be a harmonic function which satisfies $$\liminf\_{r \to \infty} \frac1{r^2} \frac{1}{|B\_r|}\int\_{B\_r} |u|^2 = 0.$$ Then $u$ is constant.
All Liouville theorems of this sort are soft/qualitative versions of harder/more quantita... | 3 | https://mathoverflow.net/users/5678 | 430233 | 174,285 |
https://mathoverflow.net/questions/430246 | 4 | Let
$$
Lu=-a\_{ij}(x)\partial\_{ij}u+b\_i(x)\partial\_i u
$$
be a uniformly elliptic operator, with $A(x)=(a\_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\Omega\subset \mathbb R^d$ is as smooth as needed (but bounded).
In one of its elementary versions, the classical [Hop... | https://mathoverflow.net/users/33741 | Nonsmooth version of Hopf boundary point lemma | I think this is just the comment following Lemma 3.4 of Gilbarg and Trudinger (specifically equation 3.11).
I should add that lowering the regularity of the boundary seems like a harder problem (and is I think false in the case of a square). GT require an interior sphere condition. L. Rosales (see for instance [Gener... | 7 | https://mathoverflow.net/users/127803 | 430250 | 174,290 |
https://mathoverflow.net/questions/430273 | 8 | Many geometries (Riemannian, symplectic, complex, Kähler, Calabi-Yau) can be defined as categories of G-structures on manifolds with the first integrability condition (zeroing of torsion of G-structure), and structure-preserving morphisms are defined naturally if $G \to GL\_n$ is monomorphism. Is it possible to define ... | https://mathoverflow.net/users/148161 | Is it possible to define contact manifolds as manifolds with a G-structure? | A contact structure on $M^{2n+1}$ defines a $G$-structure (actually, it defines more than one, but there is a 'minimal' $G$-structure that is preserved by all contact transformations, and that is the one that people usually consider). Conversely, a $G$-structure on $M^{2n+1}$ comes from a contact structure provided tha... | 11 | https://mathoverflow.net/users/13972 | 430275 | 174,295 |
https://mathoverflow.net/questions/430242 | 6 | The question is to describe ALL integer solutions to the equation in the title. Of course, polynomial parametrization of all solutions would be ideal, but answers in many other formats are possible. For example, answer to famous Markoff equation $x^2+y^2+z^2=3xyz$ is given by Markoff tree. See also this previous questi... | https://mathoverflow.net/users/89064 | What are the integer solutions to $z^2-y^2z+x^3=0$? | We get $z(y^2-z)=x^3$. Thus $z=ab^2c^3$, $y^2-z=ba^2d^3$ for certain integers $a, b, c, d$ (that is easy to see considering the prime factorization). So $ab(bc^3+ad^3)=y^2$. Denote $a=TA$, $b=TB$ (each pair $(a, b) $ corresponds to at least one triple $(A, B, T)$, but possibly to several triples). You get $T^3AB(Bc^3+A... | 7 | https://mathoverflow.net/users/4312 | 430278 | 174,296 |
https://mathoverflow.net/questions/430268 | 21 | For $\gamma$ an ordinal, let “$H\_\gamma$” be the statement:
>
> For all ordinals $\alpha$, we have $2^{\aleph\_\alpha} = \aleph\_{\alpha+\gamma}$.
>
>
>
So clearly $H\_0$ is false, and so is $H\_\omega$; in fact, $H\_\gamma$ implies that $\gamma$ is successor (because otherwise $\operatorname{cf}\gamma = \ome... | https://mathoverflow.net/users/17064 | What is known about the consistency of $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$ for all $\alpha$? | 1. By a result of Patai, $\gamma$ should be finite (this is exercise 5.15 in Jech's book).
2. For any finite $n>0, H\_n$ is consistent, see Merimovich's paper [*A power function with a fixed finite gap everywhere*](https://arxiv.org/abs/math/0005179).
---
For completeness and since not everyone has Jech's book at... | 21 | https://mathoverflow.net/users/11115 | 430293 | 174,300 |
https://mathoverflow.net/questions/430234 | 5 | Let $\mathcal{G}\_n = \{ N(\mu,\Sigma) ; \mu \in \mathbb{R}^n, \Sigma > 0\}$ be the collection of Gaussian distributions on $\mathbb{R}^n$ with full support.
If $f : \mathbb{R}^n \to \mathbb{R}^k$ is measurable, $k\leq n$, and $f \# \gamma \in \mathcal{G}\_k$ for all $\gamma \in \mathcal{G}\_n$, is it true that $f$ i... | https://mathoverflow.net/users/99418 | Gaussian-to-Gaussian transformations are affine a.e.? | The result for $k=n$ with proof may be found in Theorem 2 of
*Nabeya, Seiji; Kariya, Takeaki*, [**Transformations preserving normality and Wishart-ness**](http://dx.doi.org/10.1016/0047-259X(86)90081-3), J. Multivariate Anal. 20, 251-264 (1986). [ZBL0602.62037](https://zbmath.org/?q=an:0602.62037).
For an interesti... | 1 | https://mathoverflow.net/users/64449 | 430309 | 174,306 |
https://mathoverflow.net/questions/430301 | 3 | I would like to ask the following question.
I am searching for a reference for the following statement:
Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely generated $A$-module. Then the following assertions are equivalent.
$\bullet$ The module $M$ is absolutely indecomp... | https://mathoverflow.net/users/91107 | Reference request for equivalent formulations of being absolutely indecomposable | This is Theorem 30.29 in
*Curtis, Charles W.; Reiner, Irving*, Methods of representation theory, with applications to finite groups and orders. Vol. I, Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XXI, 819 p. \textsterling 40.70 (1981). [ZBL0469.20001](https://zbma... | 6 | https://mathoverflow.net/users/22989 | 430314 | 174,307 |
https://mathoverflow.net/questions/430156 | 2 | Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J\_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
>
> For which $X$'s is it true that $J\_A+J\_B=J\_{A\cap B}$?
>
>
>
I can prove this if $X$ is hereditary normal, as well as in the case when $C(X)$ is complete ... | https://mathoverflow.net/users/53155 | On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$ | This characterizes normality.
That it implies normality was observed above by Remy.
Conversely, assume $f$ vanishes on $A\cap B$. Define $h:A\cup B\to\mathbb{R}$ by $h(x)=f(x)$ if $x\in A$ and $h(x)=0$ if $x\in B$. By the Tieze-Urysohn theorem $h$ has a continuous extension $H:X\to\mathbb{R}$. That extension belongs ... | 7 | https://mathoverflow.net/users/5903 | 430316 | 174,308 |
https://mathoverflow.net/questions/430317 | 4 | Let $M$ be closed orientable $2n$-manifold, where $n$ is odd. It is well known that the $\mathbb Z$-module $H^\bullet(M;\mathbb Z)$ has graded-commutative multiplication and $H^{2n}(M;\mathbb Z)\simeq\mathbb Z$. So (by Poincaré duality) there is a skew-symmetric quadratic form $[-]\smile[-]$ on $H\_n(M;\mathbb Z)$.
C... | https://mathoverflow.net/users/76500 | Intersection form of $2n$-manifold for odd $n$ | The following argument can be phrased in terms of cohomology (it amounts to a proof that the Euler class of an odd-rank bundle is 2-torsion) but here is a purely intersection-theoretic phrasing.
It suffices to show that if $E \to S$ is any vector bundle with $\text{rank}(E) = \dim S = 2k+1$, then a generic section ha... | 9 | https://mathoverflow.net/users/40804 | 430319 | 174,309 |
https://mathoverflow.net/questions/430283 | 2 | I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that:
$$AX = B$$
Given that $X$ is orthonormal this is also true:
$$A = BX^T$$
**How to find $X$?**
I tried to use [Moore-Penrose inversion](https://... | https://mathoverflow.net/users/491098 | Orthonormal solution of overdetermined linear equations | You can formulate this problem as an [orthogonal Procustes problem](https://en.wikipedia.org/wiki/Orthogonal_Procrustes_problem):
$$
\min\_{X \text{ orthogonal}} \|AX-B\|\_F.
$$
With a transpose you can convert from the notation in the Wikipedia page to this form: the solution is $X=UV^T$, where $A^TB = U\Sigma V^T$ ... | 3 | https://mathoverflow.net/users/1898 | 430322 | 174,311 |
https://mathoverflow.net/questions/430302 | 2 | Let $\mathcal L$ be the space of lattices in $\mathbb R^d$. The **$i$-th successive minimum** of $L\in \mathcal L$, denoted $\lambda\_i(L)$ is the infimum of the radii of the balls containing $i$-linearly independent vectors in $L$.
Let $v\_1$ denote any vector in $L$ with $\lVert v\_1\rVert=\lambda\_1(L)$. Now proje... | https://mathoverflow.net/users/489992 | Successive minima of a lattice and projection along the the shortest nonzero vector | By taking a shortest lift from $L\_1$ to $L$ you obtain that $\lambda\_{i+1}(L)\le \lambda\_i(L\_1)+\frac{1}{2}\lambda\_1(L) \le \lambda\_i(L\_1)+\frac{1}{2}\lambda\_{i+1}(L)$. This gives $\frac{1}{2}\lambda\_{i+1}(L) \le \lambda\_i(L\_1)$, i.e. $\lambda\_{i+1}(L) \le 2\lambda\_i(L\_1)$.
| 2 | https://mathoverflow.net/users/40821 | 430323 | 174,312 |
https://mathoverflow.net/questions/430305 | 2 | I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$:
$$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$
Where $\cos(x,y)$ gives cosine of the angle between vectors $x$,$y$
$$\cos(x,y)=\frac{\langle x, y\rangle}{\|x\| \|y\|}$$
In simulations, I'm finding it be... | https://mathoverflow.net/users/7655 | Triangle equality for cosine similarity in high dimensions | $\newcommand{\R}{\mathbb R}$Here is a straightforward explanation in the case of adding iid standard normal random variables (r.v.'s). Here we have random vectors
\begin{equation}
U:=X,\quad V:=X+Y,\quad W:=X+Y+Z,
\end{equation}
where $X,Y,Z$ are independent standard normal random vectors in $\R^d$. Letting $\cdot$ d... | 2 | https://mathoverflow.net/users/36721 | 430325 | 174,313 |
https://mathoverflow.net/questions/430318 | 2 | Is there any reference to the proof of following: let $T$ denote the [Lannes functor.](https://encyclopediaofmath.org/wiki/Lannes-T-functor) Then (see the link above for more details) for any finite $E$-complex $X$ (where $E$ is finite-dimensional $\mathbb F\_p$-vector space), one should have $T\_EH\_E^\*(X) = H^\*BE \... | https://mathoverflow.net/users/477793 | Fixed points cohomology via Lannes T-functor | There is something wrong with your assertion, which, when $X$ is specialized to a point, says that $$T\_EH^\*(BE) = H^\*(BE).$$ But this is **not** true. (I can't access the Dywer-Wilkerson paper you mention right at the moment.)
Lannes' famous 1992 paper in Pub. IHES (en francais!) has a functor he calls $Fix$, and ... | 1 | https://mathoverflow.net/users/102519 | 430335 | 174,317 |
https://mathoverflow.net/questions/430332 | 4 | In the theory of Sobolev space, we have the following chain rule:
* For a uniformly Lipschitz function $F : \mathbf{R}\to \mathbf{R}$ such that $F(0)=0$,
and $u\in W^{1,1}(\mathbf{R}^n)$, then we have the following chain rule:
$\partial\_j F(u)=F'(u)\circ \partial\_ju$.
But how to define the function $F'(u)$? It se... | https://mathoverflow.net/users/166368 | Chain rule in Sobolev space | The main difficulty in the proof of the rule is to prove that $\nabla u=0$ a.e. on the set $u^{-1}(\Sigma)$, where $\Sigma$ is the set where $F$ is not differentiable; and where $\nabla u=0$ one defines the product $F'(u)\nabla u$ to be 0, irrespective of the fact that $F'(u)$ is defined or not a such points. See e.g. ... | 11 | https://mathoverflow.net/users/7294 | 430339 | 174,318 |
https://mathoverflow.net/questions/430333 | 3 | This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}\_X \rangle
$$
where $E$ is the pullback of the rank 2 tautological subbundle on $Gr(2, g/2 + 2)$.
In [1], the authors cons... | https://mathoverflow.net/users/458355 | Left adjoint for nested admissible categories | If $\mathcal{A} \subset \mathcal{T}$ is a right admissible subcategory and
$$
\mathcal{T} = \langle \mathcal{A}^\perp, \mathcal{A} \rangle
$$
is the corresponding semiorthogonal decomposition, the left mutation functor $\mathbf{L}\_{\mathcal{A}}$ is **defined** as the left adjoint functor of the embedding $\mathcal{A}^... | 3 | https://mathoverflow.net/users/4428 | 430361 | 174,325 |
https://mathoverflow.net/questions/430347 | 3 | Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
| https://mathoverflow.net/users/110301 | Regularity of lipschitz and derivable function | I claim that a function with these properties need not be $C^1$.
We start with the function $f: t \in (-1,1)\setminus \{ 0 \} \mapsto \operatorname{sin}(1/t)$, and we also set $f(0) = 0$. The antiderivative of $f$, namely $F: x \in (-1,1) \mapsto \int\_0^x \operatorname{sin}(1/t) \mathrm{d} t$ is Lipschitz, but not $... | 6 | https://mathoverflow.net/users/103792 | 430362 | 174,326 |
https://mathoverflow.net/questions/430363 | 2 | For $G = GL\_n$, it is known that the generic fibers of the Hitchin fibration are the Picard stacks of line bundles on the corresponding spectral curves and the duality of Hitchin fibrations in this case amounts to the self-duality of the Jacobian of an algebraic curve. Note however that these statements are valid on a... | https://mathoverflow.net/users/2623 | Duality of Hitchin fibrations in type A | The relevant open subset is simply the open subset where the spectral curve is smooth. Thus it is the nonvanishing locus of some discriminant polynomial.
The Hitchin fiber is the moduli space of bundles on the spectral curve, generically free of rank one, whose pushforward to the base curve is locally free of rank $n... | 6 | https://mathoverflow.net/users/18060 | 430364 | 174,327 |
https://mathoverflow.net/questions/430365 | 56 | Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? If not, is it because:
* It is not as interesting as quadratic forms?
* It is so hard that no-one has yet written about... | https://mathoverflow.net/users/173315 | There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...? | I once asked André Weil the same question.
When I was college, taking a course that discussed quadratic forms, Weil gave a guest lecture to the students about that topic. After the talk, I raised my hand and asked him why there was such a big deal in math about quadratic forms while it seemed there was nothing compar... | 71 | https://mathoverflow.net/users/3272 | 430367 | 174,329 |
https://mathoverflow.net/questions/429873 | 5 | Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map
\begin{align\*}
\pi\colon \mathcal{X}\to \Delta
\end{align\*}
and $t,t' \in \Delta$ such that $X$ and $X'$ are biholomorphic to the fibers $\pi... | https://mathoverflow.net/users/141157 | Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler? | I don't think this is known. For hyperkahler manifolds, conjecturally,
all smooth complex deformations are class C and birational to hyperkahler.
If this is true, your conjecture would follow automatically. The only
relevant publication that I am aware of is
<https://arxiv.org/abs/1703.02001>
*Arvid Perego*
Kähle... | 1 | https://mathoverflow.net/users/3377 | 430386 | 174,336 |
https://mathoverflow.net/questions/430255 | 7 | Is there a theory T such that:
* T includes all the axioms of [CZF](https://en.wikipedia.org/wiki/Constructive_set_theory).
* T includes the Idealization, Standardization, and Transfer schemas from [IST](https://en.wikipedia.org/wiki/Internal_set_theory#Formal_axioms_for_IST).
* Every axiom of T is a theorem of IST.
... | https://mathoverflow.net/users/65915 | Is there a constructive version of internal set theory? | As you have proven (by a well-known construction), one cannot expect to have full Transfer in constructive NSA. For different but related reasons, full Standardisation is off the table, though its restriction to reals seems semi-constructive. In fact, one can have Idealisation and weak versions of the other IST axioms,... | 8 | https://mathoverflow.net/users/33505 | 430387 | 174,337 |
https://mathoverflow.net/questions/363882 | 6 | The Monster group is the largest of the sporadic simple groups, and has been proven by Wilson to also be a Hurwitz group. It has a presentation in terms of Coxeter groups, specifically Y443 along with the "spider" relator, and quotienting out by the center. However, I am interested in a presentation as a Hurwitz group.... | https://mathoverflow.net/users/38744 | Presentation of the Monster as a Hurwitz group | I have computed two pairs of generators $(a,b)$ of the Monster satisfying
the relations $a^2 = b^3 = (ab)^7 = 1$ using [1]. In both cases $a$ is of
class 2B, $b$ is of class 3B in the Monster, and the commutator $[a, b]$
has order 39. This gives an upper bound for the minimal order of that commutator.
More details of t... | 8 | https://mathoverflow.net/users/105705 | 430393 | 174,340 |
https://mathoverflow.net/questions/430337 | 1 | Consider a $k$ regular graph of $n$ vertices, where $3 \leq k \leq (n-1)$. Is there any upper or lower bound, in the worst case, known for either the tree-width or the clique width of each $k$ regular family?
| https://mathoverflow.net/users/166840 | Tree width and clique width of regular graphs | For each $k$-regular family, the treewidth and cliquewidth can be both $\Theta(n)$, due to the existence of expanders.
By [On Balanced Separators, Treewidth, and Cycle Rank](https://arxiv.org/abs/1012.1344) Thm. 2.1, $tw(G) \geq \tilde s(G) -1$, and by the definition of the strict balanced separator number, $\tilde s... | 3 | https://mathoverflow.net/users/125498 | 430403 | 174,343 |
https://mathoverflow.net/questions/430336 | -1 | Let $G$ be a finite group with no abelian subfactor in its composition series.
Is $G$ obtained from simple groups by iterating semidirect products?
(Initially it was asked whether $G$ is a direct product of simple groups, but $A\_5\wr A\_5$ was mentioned as an immediate counterexample.)
| https://mathoverflow.net/users/475097 | Splitting of a finite group with no abelian subfactor in composition series | There are groups that look like wreath products, but where the base group has no complement, so they are not semidirect products. The theory is described in an old paper of mine (and probably elsewhere):
D. F. Holt, Embeddings of group extensions into Wreath products, Quar. J. Math. (Oxford) 29 (1978), 463--468.
It... | 8 | https://mathoverflow.net/users/35840 | 430413 | 174,345 |
https://mathoverflow.net/questions/415409 | 4 | Let $\mathcal{V}$ be a semi-monoidal category, meaning it satisfies the axioms of a monoidal category except missing a unit and the unit axiom. One could then still go about defining a $\mathcal{V}$-category by dropping the requirement of having unit morphisms.
One concern is that without unit morphisms there is no w... | https://mathoverflow.net/users/160045 | Enriched categories over a semi-monoidal category | Expanding upon my comment: categories without units are called [semicategories](https://ncatlab.org/nlab/show/semicategory). You can enrich a semicategory in a semigroupal category, which is what you describe. The Yoneda lemma is subtle with semicategories, but see [On regular presheaves and regular semi-categories](ht... | 4 | https://mathoverflow.net/users/152679 | 430422 | 174,348 |
https://mathoverflow.net/questions/430424 | 6 | Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.
**Question.**
(i) Are the tangent cones at the origin regular, that is $\operatorname{sing} \mathbf{C} = \{ 0 \}?$ (ii) If *one... | https://mathoverflow.net/users/103792 | What is the current status on bad tangent cones at isolated singularities? | (i) This used to be a wide open area, but recently there has been some progress: Gabor Székelyhidi has constructed an example of an isolated singularity with a cylindrical tangent cone here: <https://arxiv.org/pdf/2107.14786.pdf> . If you are willing to change the metric from the flat metric to some other Riemannian me... | 6 | https://mathoverflow.net/users/1540 | 430437 | 174,354 |
https://mathoverflow.net/questions/430434 | 2 | In Angella and Tomassini's [paper](https://link.springer.com/article/10.1007/s00222-012-0406-3) p.75, there is an exact sequence:
>
> $\cdots\to B^{\bullet,\bullet}\to H\_{\bar\partial}^{\bullet,\bullet}\to H\_A^{\bullet,\bullet}\to \cdots$
>
>
>
where $B^{\bullet,\bullet}:=\frac{\ker\bar\partial\cap\text{im }... | https://mathoverflow.net/users/99826 | The kernel of $H^{\bullet,\bullet}_{\bar\partial}(X)\to H_A^{\bullet,\bullet}(X)$ | Let $\alpha$ be a $\bar{\partial}$-closed form. Denote its Dolbeault cohomology class by $[\alpha]\_{\bar{\partial}}$ and its Aeppli cohomology class by $[\alpha]\_A$; note that the map $g$ is given by $g([\alpha]\_{\bar{\partial}}) = [\alpha]\_A$. Likewise, if $\alpha' \in \ker\bar{\partial}\cap\operatorname{im}\parti... | 5 | https://mathoverflow.net/users/21564 | 430439 | 174,355 |
https://mathoverflow.net/questions/430391 | 5 | Background: A referee has suggested a shorter proof of one of my results, but I'm having trouble justifying one of their assertions. The setting is that $A$ is a commutative ring, and the referee's suggestion is to reduce to the case that $A$ is local and finite (as a set) in the following way:
* First assume that $A... | https://mathoverflow.net/users/1474 | For a finite-type $\mathbb{Z}$-algebra $A$, is the intersection of all ideals $I$ such that $A/I$ is finite and local necessarily zero? | A proof of the proposed result that is similar (if not identical) to [Peter Kropholler's](https://mathoverflow.net/a/430392/84349) and YCor's proofs can be derived from two *well-known* results, namely **[Lemma 2](https://mathoverflow.net/questions/57515/a-finitely-generated-mathbbz-algebra-that-is-a-field-has-to-be-fi... | 2 | https://mathoverflow.net/users/84349 | 430446 | 174,357 |
https://mathoverflow.net/questions/430445 | 1 | Let $X$ be a smooth projective surface. Let $p$ be a closed point of $X$. Let $k(p)$ be the corresponding skyscraper sheaf, then actually we could use Grothendieck-Riemann-Roch to calculate the Chern of the coherent sheaf $k(p)$.
However, now let $U\cong \mathrm{Spec}A$ be a open subset containing $p$, $m$ be the max... | https://mathoverflow.net/users/158876 | Chern class of torsion sheaf support on a point | If you can figure out the Chern character of $i\_\*(\mathcal{O}\_Z)$ where $Z = \text{Spec}(A/m)\hookrightarrow X$ by GRR then you use the following short exact sequence:
$$0 \rightarrow m/m^2 \rightarrow \mathcal{O}\_Y\rightarrow \mathcal{O}\_Z\rightarrow 0$$
Note the pushforward is also going to be exact. The ker... | 3 | https://mathoverflow.net/users/127776 | 430447 | 174,358 |
https://mathoverflow.net/questions/430450 | 3 | This [paper](https://reader.elsevier.com/reader/sd/pii/0167715294901104?token=A61379C3631ADCAC256F64615D513FEE297A7F515D6873C962AA17DFBF8B8992D4BBB4DE58A9018CA4D09009E587C0E9&originRegion=us-east-1&originCreation=20220914163053) proves a probabilistic version of Taylor's theorem
\begin{equation\*}
\mathbb{E}g(X) = \s... | https://mathoverflow.net/users/159841 | Probabilistic Taylor theorem for concave functions | If $g^{(4)}\le0$, [then](https://en.wikipedia.org/wiki/Taylor%27s_theorem#Explicit_formulas_for_the_remainder)
$$g(x)=\sum\_{k=0}^3\frac{g^{(k)}(0)}{k!}\,x^k+\frac{x^4}4\,
\int\_0^1g^{(4)}(sx)(1-s)^3\,ds
\le\sum\_{k=0}^3\frac{g^{(k)}(0)}{k!}\,x^k$$
for real $x$.
Replacing here $x$ by $X-\mu$ and assuming that $E|X|^3... | 3 | https://mathoverflow.net/users/36721 | 430456 | 174,362 |
https://mathoverflow.net/questions/430440 | 13 | Let $f\in C([0,1],[0,1])$ be such that:
$$\forall x\in [0,1], \; \exists k\in \mathbb N, \; f^{\circ k}(x)=0.$$
Is it true that $f$ is nilpotent (i.e., that there is some $k$ such that $f^{\circ k}=0$)?
Here $f^{\circ k}$ denotes the $k$th iterate of $f$.
| https://mathoverflow.net/users/110301 | Does locally nilpotent imply nilpotent for continuous self-maps of intervals? | Yes, this implies that $f$ is nilpotent.
As explained in my comment, $f(0)=0$ because otherwise Sarkovski's theorem would give us other periodic orbits which, of course, won't visit $x=0$. We also know that $f(x)<x$ for $x>0$.
Decompose the open set $\{x: f(x)>0\}=\bigcup I\_n$ into its connected components. Clearl... | 7 | https://mathoverflow.net/users/48839 | 430458 | 174,363 |
https://mathoverflow.net/questions/430419 | 7 | I'm a science/math journalist [ger] and currently I'm working on an article about prize money for open problems (Millennium Prize Problems and such). One section will be about the history of prize money and rewards. While I know about the culture of betting and rewards for open problems in the Lwow school and Erdös' fa... | https://mathoverflow.net/users/479777 | Best sources for the history of prize money for open mathematical problems | Here is a long list of [Mathematics awards](https://en.wikipedia.org/wiki/List_of_mathematics_awards) and many of them have a prize money too (you can click on each of them in order to read more and find winners data).
Another couple of important prizes to mention (without taking into account the Clay Mathematics In... | 3 | https://mathoverflow.net/users/481829 | 430472 | 174,368 |
https://mathoverflow.net/questions/430401 | 3 | (The construction of matrix $\mathbf{A}$ is not difficult to be understood. You can first jump to **A Toy Example** to take a glance. Any idea or suggestion would be appealing for me.)
---
**The Original Problem:**
Given $N,D\in\mathbb{Z}^+~(D\ge N)$ and $\alpha\in\mathbb{R}^+$, the vector $\mathbf{p}$ and the ... | https://mathoverflow.net/users/490652 | $\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$ | **Q1** The determinant is $\prod\_{n=1}^{N-1} (1 - e^{-2\alpha(p\_{n+1}-p\_n)})$.
**Q2** Yes, using the answer to **Q1**.
**Q3** Yes, using the answer to **Q1**.
---
---
The formula for **Q1** is proved by induction on $N$.
The base case $N=1$ is just $\det(1)\_{1\times1} = 1$.
For $N>1$, let $c = e^{-\al... | 5 | https://mathoverflow.net/users/14830 | 430486 | 174,370 |
https://mathoverflow.net/questions/430355 | 30 | Starting with a single stick of unit length, a point $p \in (0, 1)$ is picked uniformly at random along the stick and the stick is snapped, producing two sticks of length $p$ and $1-p$.
At each next stage, a stick is picked uniformly at random, and a point is picked uniformly at random along the length of that stick,... | https://mathoverflow.net/users/173490 | Expected length of longest stick in a stick snapping process | It seems that the length of the longest stick is of order $n^{2\sqrt{2}-3} = n^{-0.171\ldots}$ as $n\to\infty$. This follows from a discrete-time analogue of the homogeneous fragmentation process, see chapter 1.5 of J. Bertoin, *Random fragmentation and coagulation processes*. Vol. 102. Cambridge University Press, 2006... | 18 | https://mathoverflow.net/users/47484 | 430495 | 174,373 |
https://mathoverflow.net/questions/430492 | 7 | Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}\_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{F}\_q\}$$ is the whole of $\mathbb{F}\_q$, unless $q$ is small (with respect to $n$).
This does make sense because if... | https://mathoverflow.net/users/45242 | Sum of two $n$th powers in finite fields | Yes, this is true, and is proved e.g. as Corollary 3 of Small's "[Diagonal equations over large finite fields](https://doi.org/10.4153/CJM-1984-016-6)" (Can. J. Math. 1984).
Small actually gives explicit bounds on how large $q$ needs to be in terms of $n$ — in particular the equation $ax^n+by^n$ generates all of $\ma... | 11 | https://mathoverflow.net/users/385 | 430496 | 174,374 |
https://mathoverflow.net/questions/430488 | 13 | Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an example of a reflexive Banach space $X$ whose unit ball does not equal the convex hull of its extreme points? Such an exampl... | https://mathoverflow.net/users/58366 | Is there a reflexive Banach space whose ball is not the convex hull of its extreme points? | I'd try $X:=\ell\_2$ with an equivalent but non strictly convex norm.
Let $(e\_k)\_{k\ge0}$ be the standard Hilbert basis of $\ell\_2$. Consider the sets:
* $A:=\{0\}\cup\{2^{-k}e\_k:k\ge 1\}$,
* $B$, the closed unit ball of $\ell\_2$,
* $C:=\overline{\text{co}}\, A$, a compact subset of the hyperplane $e\_0^\perp$... | 7 | https://mathoverflow.net/users/6101 | 430502 | 174,376 |
https://mathoverflow.net/questions/430462 | 1 | Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have $E[\prod\_{i=1}^n Z\_i^2] = 1$ by independence.
Is there a method for bounding such functions if $X$ was an $n$ dimen... | https://mathoverflow.net/users/491250 | Comparison of Rademacher and Gaussian moments under linear transformations | By the arithmetic mean--geometric mean inequality and the condition $Z=U^\top X$ for an orthogonal matrix $U$,
$$\prod\_1^n Z\_i^2\le\Big(\frac1n\sum\_1^n Z\_i^2\Big)^n
=\Big(\frac1n\sum\_1^n X\_i^2\Big)^n=1,$$
since $X\_i=\pm1$ for all $i$. So,
$$E\prod\_1^n Z\_i^2\le1,$$
as desired.
| 2 | https://mathoverflow.net/users/36721 | 430508 | 174,379 |
https://mathoverflow.net/questions/430518 | 1 | As we know we can construct unitary matrix as $H=H\_1H\_2\dots,H\_k$, by stacking householder matrices $H\_i\in \mathbb{R}^{d\times d}$. The number of householder matrices we use, i.e., $k$, determines the expressivity of construction.
Conversely, for a given unitary matrix $H$, how can I know how many householder ma... | https://mathoverflow.net/users/127318 | How many householder matrices do I need for constructing a given unitary matrix? | A Householder decomposition of a $d \times d$ unitary $H$ can be achieved with *at most* $d$ Householder matrices (see [Theorem 1](https://core.ac.uk/download/pdf/82680205.pdf)) and a tighter upper bound is $d-m$ Householder matrices where $m = \dim( \operatorname{ker}(H-I\_d) )$ (see [Theorem 2](https://core.ac.uk/dow... | 3 | https://mathoverflow.net/users/64449 | 430525 | 174,384 |
https://mathoverflow.net/questions/430433 | 3 | I have a few related questions. First I would appreciate it if someone could provide me with a reference for the following
"Complex unitary irreducibles of virtually abelian groups have bounded degrees."
I am only concerned with complex unitary representations in this question.
Do we have a quantitative version o... | https://mathoverflow.net/users/491235 | Irreducibles of virtually abelian finitely generated groups | I can at least explain how this bounded degree property works roughly when $G$ is a countable virtually abelian group. Kaplansky shows in his paper Groups with representations of bounded degree, Canadian J. Math vol. 1 (1949), 105-112 that the group ring of a virtually abelian group satisfies a polynomial identity.
I... | 4 | https://mathoverflow.net/users/15934 | 430535 | 174,386 |
https://mathoverflow.net/questions/430530 | 9 | I have already asked this at [MSE](https://math.stackexchange.com/questions/4524317/propagators-and-pdes) but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. Mathematically, these derivations are somewhat magical (ty... | https://mathoverflow.net/users/89014 | Propagators and PDEs | You'll find some more info about the fundamental solutions of the wave equation in chapter 5.D of [Folland](https://press.princeton.edu/books/hardcover/9780691043616/introduction-to-partial-differential-equations) and chapter I.7 of [Trèves](https://store.doverpublications.com/0486453464.html).
The trick you use is the... | 9 | https://mathoverflow.net/users/49417 | 430536 | 174,387 |
https://mathoverflow.net/questions/430532 | 1 | Here are two different versions of Gaussian ballot theorems in the literature, each on different while similar events but the rate is quite different:
1. [P39, Probability Result 1](http://wrap.warwick.ac.uk/129549/7/WRAP-moments-functions-squareroot-cancellation-chaos-Harper-2020.pdf): For any independent sequence o... | https://mathoverflow.net/users/174600 | What happens in the difference rate between these two versions of ballot theorem? | $\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$The paper linked in your post contains a reference to a reference to an apparent proof, which I hope results in an overall valid proof.
Anyhow, we can assess the plausibility of the relations
\begin{equation\*}
P(S\_j\le a\ \forall j\le ... | 2 | https://mathoverflow.net/users/36721 | 430548 | 174,390 |
https://mathoverflow.net/questions/430545 | -1 | Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\alpha\gamma a^3=0$$
$$(-\alpha\gamma+2\gamma\delta)a^2b+(\delta^2-(\alpha\delta+\beta\gamma))ab^2-\beta\delta b^3+\gamma^2a^3=... | https://mathoverflow.net/users/10035 | Does this quadratic system admit an integral or a rational solution? | In my comments I employed Maple, which uses tools like Grobner bases to solve polynomial equations. But now I'll try to do it by hand. Let $E\_1,E\_2,E\_3$ be the three equations. A rational solution of $\{E\_1,E\_2,E\_3\}$ can be turned into an integer solution of $\{E\_2,E\_3\}$ by multiplying by a common denominator... | 2 | https://mathoverflow.net/users/9025 | 430559 | 174,393 |
https://mathoverflow.net/questions/430543 | 6 | $\DeclareMathOperator\Int{Int}\DeclareMathOperator\Ext{Ext}$Suppose $E\subset\mathbb{R}^n$ is a set of finite perimeter and suppose that the measure theoretic boundary $\partial^\*E=\mathbb{R}^n\setminus(\Int(E)\cup \Ext(E))$ is closed, where $\Int(E)=\{x\in\mathbb{R}^n\,:\, \lim\_{r\to 0}\frac{\mathscr{L}^n(E\cap B\_r... | https://mathoverflow.net/users/351083 | If the measure theoretic boundary is closed must it coincide with the topological boundary? | This is indeed true; you can find the answer to your question in Francesco Maggi's book for example, by combining Proposition 12.9 and Remark 15.3. I'll paraphrase the argument.
Let $E \subset \mathbf{R}^n$ be a Caccioppoli set, and $\mu\_E$ be its Gauss–Green measure. Then $\partial^\* E \subset \operatorname{spt} \... | 4 | https://mathoverflow.net/users/103792 | 430577 | 174,399 |
https://mathoverflow.net/questions/336219 | 0 | I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$
$$\bar{S}(k)=\sum\_{p+q=k}[p]^{2m+1}q $$
The box symbol over $p$ denote that when p=0 it should be treated as $[0]=1$
I have seen the de... | https://mathoverflow.net/users/45170 | If the coefficient of the polynomial positive | When $m=1$ and $k$ is odd, I compute that $\bar{S}(k) = (3k^5-20k^3+17k)/120$, which has a negative coefficient.
| 1 | https://mathoverflow.net/users/2807 | 430586 | 174,401 |
https://mathoverflow.net/questions/430547 | 3 | In [this paper](https://arxiv.org/abs/2003.10470), Danny Calegari shows that taut foliations in (let's say closed for simplicity) 3-manifolds are precisely those which admit a map $f: M \to S^2$ which restricts to a branched cover on each leaf. I'm trying to unpack what this means for the topology of $M$.
The TL;DR i... | https://mathoverflow.net/users/314845 | Do taut foliations leafwise branch covering S^2 yield foliations by circles? | To see what is going on, I think it is helpful to carefully work out an explicit example. In Section 2.1 of Calegari's paper, he explains how to deal with 3-manifolds that fiber over the circle. Here is a brief description of what he does.
I will start by setting up notation (and I won't follow Calegari here since I ... | 3 | https://mathoverflow.net/users/317 | 430587 | 174,402 |
https://mathoverflow.net/questions/429842 | 13 | The physicist [Yoichiro Nambu](https://en.wikipedia.org/wiki/Yoichiro_Nambu) introduced in a 1950 paper [A Note on the Eigenvalue Problem in Crystal Statistics](http://www.totoha.net/archiv/3675.pdf) the notion of an "*eigenoperator*" (page 12, see [Nambu and the Ising model](https://arxiv.org/abs/2209.01122) for a rec... | https://mathoverflow.net/users/11260 | Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature? | Such an $X$ is an *eigenvector of $\,\operatorname{ad}(H)$*. Joint eigenspace decompositions of several $\operatorname{ad}(H\_i)$ are commonplace in math since the work of Lie, Killing, Cartan, with the joint eigenvectors called *root vectors*. So I would say that the notion “had a place” already before Nambu.
| 10 | https://mathoverflow.net/users/19276 | 430589 | 174,403 |
https://mathoverflow.net/questions/430596 | 9 | My first idea on how to do this would be:
1. $0\in\mathbf Z$
2. $\forall x\in\mathbf Z,Sx\in\mathbf Z\land Px\in\mathbf Z$
3. $P$ and $S$ are injective
4. $\forall x\in\mathbf Z,PSx=SPx=x$
5. some induction axiom that holds for decreasing sequences as well (intuitively, if $\phi$ holds for $x\_0\in\mathbf Z$ and $\fo... | https://mathoverflow.net/users/nan | What is the canonical way to extend Peano's axioms to the set of all integers? | The canonical way to extend Peano arithmetic to the integers is not by changing the language or axioms, but instead by treating integers as equivalence classes of pairs of natural numbers.
We think of an integer $x$ as any pair $(a,b)$ of natural numbers with $a-b=x$. More formally, we translate arithmetic statements... | 21 | https://mathoverflow.net/users/nan | 430600 | 174,407 |
https://mathoverflow.net/questions/430606 | 3 | Let $\Gamma=\left<\mathcal{S}\,|\,\mathcal{R}\right>$ be a group defined by the presentation, where each relator $r\in\mathcal{R}$ is a reduced word of length 3 consisting of three different symbols only in $\mathcal{S}$, not in $\mathcal{S}^{-1}$.
Take a proper subset $\mathcal{S'}$ of $\mathcal{S}$, and let $\Gamma... | https://mathoverflow.net/users/490332 | Subgroup of a group with length 3 relators | Here is an explicit example showing that assumptions (C1) and (C2) are not sufficient for your question. Let
$$G = \langle a,b,c,d \mid abc, bdc, dac, adb \rangle.$$
A quick computer calculation shows that $|G|=24$ and in fact $G \cong {\rm SL}(2,3)$ with generator images
$$ \left( \begin{array}{cc}1&2\\1&0\end{array} ... | 5 | https://mathoverflow.net/users/35840 | 430621 | 174,411 |
https://mathoverflow.net/questions/430591 | 1 | Assume you have $n$ independent binary variables $\{x\_1,\dots,x\_n\}$ and for each variable $x\_i$ you know that its value is equal to $1$ with a probability $p\_i$. I would like to enumerate the joint assignments of these variables in such a way that in $k$ steps I capture as much volume of the joint probability mass... | https://mathoverflow.net/users/491356 | Choosing $k$ different assignments of binary variables in order to capture the largest volume of the joint probability distribution | You want to find a $k$-subset $S\subseteq\{0,1\}^n$ to maximize
$$\sum\_{(y\_1,\dots,y\_n)\in S} \prod\_{i=1}^n \left(p\_{i}y\_i+(1-p\_i)(1-y\_i)\right).$$
Equivalently, find the $k$ largest (with multiplicity)
$$\prod\_{i=1}^n \left(p\_{i}y\_i+(1-p\_i)(1-y\_i)\right).$$
Equivalently, find the $k$ largest (with mul... | 0 | https://mathoverflow.net/users/141766 | 430651 | 174,421 |
https://mathoverflow.net/questions/430646 | 2 | I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet to figure out the exact subfield that would suit me best. I would like to know the best and quickest way to get a brief ... | https://mathoverflow.net/users/124001 | Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology | Algebraic topology is a research area where often even the statements (and certainly the proofs) of active research areas require a lot of background to understand. For that reason, the answer to the linked stack exchange question from 2012 is really spot on. The best way to decide which area you might want to research... | 3 | https://mathoverflow.net/users/11540 | 430659 | 174,424 |
https://mathoverflow.net/questions/430552 | 0 | [Legendre's formula](https://mathworld.wolfram.com/LegendresFormula.html) can be very easily be generalised as mentioned [here](https://projecteuler.net/action=redirect;post_id=111677) (visible after login) which is like this
${\pi}(v,p)={\pi}(v,p-1)-1.[{\pi}(v/p,p-1)-{\pi}(p-1,p-1)]$
${ \big\downarrow}$
$S(v,p)=... | https://mathoverflow.net/users/483720 | How can I convert Meissel's/Lehmer's formula for prime counting to get sum of primes | This might be a "shameless plug" but i did recently "generalize Meissel-Lehmer to count sum of the powers of primes". This was supposed to give an exposition, so it might be of help
<https://arxiv.org/abs/2111.15545>
| 2 | https://mathoverflow.net/users/483197 | 430663 | 174,426 |
https://mathoverflow.net/questions/430665 | 4 | Let $G$ be an almost $k$-simple group that is also simply connected (so that $G(k)^{+}=G(k)$). For opposite parabolic subgroups $P$ and $P^{-}$, it is known that $G(k)^{+}$ is generated by the unipotent radicals $R\_u(P)(k)$ and $R\_u(P^{-})(k)$ (Prop 1.5.4 in Margulis's book "Discrete Subgroups of Semisimple Lie group... | https://mathoverflow.net/users/15311 | Bounded generation of group by unipotent radicals of opposite parabolic subgroups | I hope you will permit me to write $P^+$ in place of $P$. Put $U^\pm = R\_u(P^\pm)$.
Yes, at least in the split case.
Suppose first that $P^+$ and $P^-$ are minimal. Put $T = P \cap P^-$. By working in $\operatorname{SL}\_2(k)$ or $\operatorname{PGL}\_2(k)$, you see that, for all $t \in k^\times$ and all roots $\al... | 6 | https://mathoverflow.net/users/2383 | 430666 | 174,428 |
https://mathoverflow.net/questions/428326 | 4 | I believe this should be a well known result, but I wasn’t able to prove or find a good reference for it.
Let $E$ and $F$ be $n$-regular, respectively $m$-regular vector bundles in the sense of Castelnuovo-Mumford, the is it true that $E\otimes F$ is at most $n+m$-regular?
The result is true for modules, let’s say ... | https://mathoverflow.net/users/43027 | Castelnuovo-Mumford regularity for tensor products of vector bundles | Yes, furthermore, your statement holds even if $E$ is locally free and $F$ is coherent.
You can apply the following fact to prove it:
>
> Suppose a coherent sheaf $\mathscr F$ on $\mathbf P$ is resolved by a
> long exact sequence $$\cdots \rightarrow \mathscr F\_2\rightarrow
> \mathscr F\_1\rightarrow \mathscr F... | 2 | https://mathoverflow.net/users/491416 | 430671 | 174,430 |
https://mathoverflow.net/questions/430627 | 5 | I have been looking into the Fermat quintic equation $a^5+b^5+c^5+d^5=0$. To exclude the trivial cases (e.g. $c=-a,d=-b$), I will take $a+b+c+d$ to be nonzero for the rest of the question. It can be shown that if $a+b+c+d=0$, then the only solutions in the rational numbers (or even the real numbers) are trivial.
I ha... | https://mathoverflow.net/users/38744 | Small Galois group solution to Fermat quintic | I have answers to your first two questions, and some insight into the third.
First, there is a quartic in the family above with Galois group contained in $A\_{4}$. One example is found by taking $q = 1/5$ and $r = 0$. This gives $5x^{4} - 5x^{3} + x^{2} + 1/5$, which has Galois group $A\_{4}$. A quartic has Galois gr... | 2 | https://mathoverflow.net/users/48142 | 430677 | 174,433 |
https://mathoverflow.net/questions/430616 | 3 | Let $A$ be a set of $n$ elements. Let $S\_1,\dots,S\_n$ be independent $k$-element random subsets of $A$. What is the probability that $S\_1,\dots, S\_n$ evenly cover $A$, i.e. each element of $A$ belongs to exactly $k$ random subsets?
| https://mathoverflow.net/users/97209 | Random covering of a set | It follows immediately from the special case (with $m=n$ and $s=t=k$) of Theorem 1 of [Canfield and Mckay](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29) that the probability -- say $p\_{n,k}$ -- in question is
$$\sim\frac1{\sqrt e}\dfrac{\displaystyle{\binom{n}{k}^n}}{\displaystyle{\binom{n^2}... | 4 | https://mathoverflow.net/users/36721 | 430683 | 174,436 |
https://mathoverflow.net/questions/430631 | 3 | Let $W$ be a standard Brownian motion, and $\mathcal F\_t$ it’s natural filtration. Let $H$ be a continuous process, adapted to $\mathcal F\_t$ and integrable with respect to $W$.
**Question:** Is it true that for all a.s. finite $\mathcal F\_t$-stopping times $\sigma$ we have
$$\lim\_{h \to 0+} \frac{1}{W\_{\sigma... | https://mathoverflow.net/users/173490 | Lebesgue differentiation theorem at a stopping time | Yes. Let $h \in (0,1)$, $Q\_h = \frac{1}{W\_{\sigma + h} - W\_{(\sigma - h) \vee 0}} \int\_{(\sigma - h)\vee 0}^{\sigma + h} (H\_s - H\_{(\sigma - h) \vee 0}) \, dW\_s$ and set $M\_h = \sqrt{-\log(h)}$. Because $H$ is continuous, and hence, $\lim\_{h \to 0} H\_{(\sigma - h) \vee 0} = H\_{\sigma} $ almost surely, the cl... | 4 | https://mathoverflow.net/users/64449 | 430684 | 174,437 |
https://mathoverflow.net/questions/430571 | 3 | **Motivation/Hand-Wavy Question:**
In [this post](https://mathoverflow.net/questions/335736/low-degree-polynomial-approximation-of-the-piecewise-linear-function-x-mapsto), it was asked what the best local approximation of $f(x):=\max\{0,x\}$ is by a *polynomial* of a given degree; with the answer provided by Chebyshev'... | https://mathoverflow.net/users/36886 | Smooth approximation of the $\max\{0,x\}$ function with controlled derivatives | Let us consider $[-1,1]$ instead.
**Lemma 1**. For any $\epsilon>0$, there exists $f\_\epsilon\in C^{\infty}\left(\left[0,1\right]\right)$,
a bijection from $\left[0,1\right]$ onto itself, such that
\begin{align\*}
f\_\epsilon(0) & =0,\\
f\_\epsilon (1) & =1,\\
\forall k\in\mathbb{N},k\geq1,f\_\epsilon^{(k)}\left(0\r... | 2 | https://mathoverflow.net/users/40120 | 430694 | 174,442 |
https://mathoverflow.net/questions/430674 | 4 | $\DeclareMathOperator\RoT{RoT}$I'm interested in the following ring. Fix a (Noetherian?) base ring $R$, and consider the category of finitely generated projective $R$-modules equipped with endomorphisms up to isomorphism. Since $\otimes$ distributes over $\oplus$, you can view this as a commutative semiring with those ... | https://mathoverflow.net/users/141571 | "Ring of traces" over a ring R | $\newcommand\HH{\mathit{HH}}$I write $[V,f]$ for the class of $(V,f)$ in your ring.
Let me prove the following property of $[V,f]$ : if $f: V\to W, g: W\to V$, then $[V,gf] = [W,fg]$.
Indeed, $[V,gf] = [V\oplus W, (gf, 0)]$, because $(W,0)$ is a commutator.
Further, by your trick about sums, $[P,-h] = -[P,h]$ so ... | 6 | https://mathoverflow.net/users/102343 | 430700 | 174,443 |
https://mathoverflow.net/questions/430396 | 6 | Suppose $G \subset \text{Iso}(M)$ is a Lie group acting smoothly on a (pseudo-)Riemannian manifold $(M, g)$. Then $G$ induces a Lie algebra of Killing vectors on $M$. In [this](https://www.semanticscholar.org/paper/Einstein-Tensor-and-3%E2%80%90Parameter-Groups-of-with-Goenner-Stachel/9917663943e0aef06647d2dabe58916673... | https://mathoverflow.net/users/491183 | Does the isometry group determine the Riemannian metric? | I think that you are missing some hypotheses on the action of $G$, otherwise there are trivial counterexamples. For example, if $G\subset\mathrm{Iso}(M,g)$ is the trivial group, then one clearly cannot reconstruct $g$ from just knowing $G$ as a subgroup of $\mathrm{Diff}(M)$. Beyond that, the most you can hope for is t... | 5 | https://mathoverflow.net/users/13972 | 430707 | 174,445 |
https://mathoverflow.net/questions/430708 | 2 | Let $\Omega=[0,1]\times [0,1]$ be the square. We say a function $f\in H^1(\Omega)$ is periodic on $\Omega$ if $f(x,0)=f(x,1)$ and $f(0,y)=f(1,y)$ (in the sense of traces of course). Now consider the problem
$$\begin{cases}-\Delta u+au=f,\\ u,u\_x,u\_y\text{ are periodic on }\Omega.\end{cases} $$
It is easily seen that ... | https://mathoverflow.net/users/345667 | Representing solutions of $-\Delta u+au=f$ when $a\leq 0$ | Take the eigenbasis $\phi\_n$ you exhibited for $a=0$. It is still an eigenbasis for $-\Delta + a$, with shifted eigenvalues $\lambda\_n + a$.
Write
$$
E\_a=\{n\in\mathbb N : \lambda\_n =- a\},
$$
which is either an empty set or a finite set. If $E\_a=\emptyset$ (case 1.) then
$$
u = \sum\_{n\in\mathbb N} \frac{1}{\la... | 1 | https://mathoverflow.net/users/40120 | 430711 | 174,446 |
https://mathoverflow.net/questions/430476 | 3 | Let $\operatorname{wt}(n)$ be [A000120](https://oeis.org/A000120), i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $a(n,m)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)=m$. In other words, $a(n,m)$ is the $n$-th number with binary weight equals $m$.
I conjecture... | https://mathoverflow.net/users/231922 | Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$ | Notice that for $i\in\{0,1,\dots,2^{n-1}+n\}$ we have
$$a(i+1,2^{n-1}+n) = 2^{2^{n-1}+n+1} - 1 - 2^{2^{n-1}+n-i}.$$
Then the sum in question can be easily computed:
\begin{split}
& a(1,2^{n-1}+n)+\sum\_{i=1}^{2^{n-1}}a(i+2,2^{n-1}+n)\\
&= \sum\_{i=0}^{2^{n-1}+1} a(i+1,2^{n-1}+n) - (2^{2^{n-1}+n+1} - 1 - 2^{2^{n-1}+n-... | 2 | https://mathoverflow.net/users/7076 | 430713 | 174,448 |
https://mathoverflow.net/questions/430657 | 2 | Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert\_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle\_{H})$ be a separable Hilbert space densely embedded in $\Theta$, and let $\mu$ be a Gaussian measure on $\Theta$ with characteristic functional $\exp... | https://mathoverflow.net/users/117692 | $\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space? | The answer to the question is no, even in the case $d=1$. Take for example $H = \ell^2$, $\Theta = \{\xi\,:\, \|\xi\| = \sup\_{n\ge 1} |\xi\_n|/(1+\log n) < \infty\}$, $\mu$ the law of i.i.d. normals, and $\Psi(\xi) = \sum\_n \xi\_n/n^{3/4}$. Assume by contradiction that a version of $\Psi$ exists that is continuous at... | 4 | https://mathoverflow.net/users/38566 | 430714 | 174,449 |
https://mathoverflow.net/questions/430705 | 4 | Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int\_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not know how to use the assumption that $ f $ is of bounded variation. Can you give me some hints?
| https://mathoverflow.net/users/241460 | The decay of Fourier coefficients and the continuity of functions | The (distributional) derivative $\mu=f'$ is a measure, and by assumption $\widehat{\mu\_n}=o(1)$. By Wiener's theorem, this implies that $\mu$ does not have a point part, so $\mu$ is a continuous measure and thus $f$ is a continuous function. Compare Corollary 13.11 of my lecture notes [here](https://math.ou.edu/%7Ecre... | 4 | https://mathoverflow.net/users/48839 | 430715 | 174,450 |
https://mathoverflow.net/questions/430581 | 0 | Let $(X,d)$ be a complete separable geodesic *(thus connected and path connected)* metric space of non-positive curvature (in the sense of [Ballmann](https://people.mpim-bonn.mpg.de/hwbllmnn/archiv/NPC0606.pdf#page=3)) and fix some $x\_0\in X$. Let $C\_0(I,X)$ denote the set of continuous functions $f:[0,1]\rightarrow ... | https://mathoverflow.net/users/491352 | Curvature of space of paths into a NPC space | Maybe I'm missing something. But it looks like if you take $X = \mathbb{R}$ than $C\_0(I, X)$ is a set of continuous functions on $[0,1]$ with $0$ at $0$. And than it's universal aka (almost) contains an isometric copy of every finite metric space. And NPC is very restrictive in terms of what $4$-point subsets are allo... | 1 | https://mathoverflow.net/users/32454 | 430723 | 174,453 |
https://mathoverflow.net/questions/430726 | 6 | **"Real-life" motivation.** The German satirical magazine *Der Postillon* [suggested](https://www.der-postillon.com/2016/08/7-wege-rauchen.html) a few measures for deterring smokers from their bad habit. I especially liked the idea of inserting one "prank cigarette" per pack, giving the smoker a reminder in form of a (... | https://mathoverflow.net/users/8628 | Expected maximum number of "prank cigarettes" in an average pack | $\newcommand{\Si}{\Sigma}$Let $X\_n$ be the maximum number of prank cigarettes any pack receives, so that $M\_n=EX\_n$. Note that $X\_n=\max(N\_1,\dots,N\_n)$, where $(N\_1,\dots,N\_n)$ has the multinomial distribution with parameters $n;\frac1n,\dots,\frac1n$. So, by Theorem 1 or formula (6) of [Raab and Steger](https... | 10 | https://mathoverflow.net/users/36721 | 430731 | 174,456 |
https://mathoverflow.net/questions/430669 | 6 | Let $(\mathbb{S}^2,g)$ be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and have the same length $L > 0$, say.
**Question 1**: in the celebrated book of Besse entitled "Manifolds all of whose geode... | https://mathoverflow.net/users/85934 | On properties of Besse spheres | It so happens I recently read this same passage in Besse's book. I am no expert, but here is how I understand it.
**Question 1.** Let $p,q \in \mathbf{S}^2$ be two arbitrary points, and $\eta: [0,L] \to \mathbf{S}^2$ be a closed geodesic containing both points, fixing $\eta(0) = p$ for example. Going along $\eta$ eit... | 2 | https://mathoverflow.net/users/103792 | 430733 | 174,457 |
https://mathoverflow.net/questions/390478 | 4 | What is the Hausdorff dimension of the subset $S\_c \subset [0,1]$ of points such that the [critical exponent](https://en.wikipedia.org/wiki/Critical_exponent_of_a_word) of their binary expansion is $c$? It's clear that $\dim\_H S\_{\infty}=1$, but what can be said for $c<\infty$?. Also, what is $\dim\_H \cup\_{c\in [2... | https://mathoverflow.net/users/167834 | Hausdorff dimension and critical exponent of words | Here are a few observations: (the last one is maybe the main point of interest)
$\bullet$ Your $S\_c$ is a closed subset of $[0,1]$ invariant under $x \mapsto 2x \pmod 1$ (usually there's a subtlety about points with multiple expansions, but none of those are in $S\_c$ for $c < \infty$), and so I think a result of Fu... | 6 | https://mathoverflow.net/users/116357 | 430734 | 174,458 |
https://mathoverflow.net/questions/429702 | 5 | This is a follow-up to the question of Joseph O'Rourke [Which metric spaces have this superposition property?](https://mathoverflow.net/q/118008/)
A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any distance-preserving map $A\to X$ can be extended to an isometry $X\to X$.
*Class... | https://mathoverflow.net/users/1441 | All-set-homogeneous spaces | This is more of an observation and a long comment than an answer.
The observation is that a fairly standard application of the Erdős–Rado theorem implies that in any metric space with more than $2^{2^{\aleph\_0}}$ many elements, there is an infinite sequence $(a\_i)\_{i<\omega}$ of distinct elements satisfying $d(a\_... | 2 | https://mathoverflow.net/users/83901 | 430738 | 174,459 |
https://mathoverflow.net/questions/430742 | 7 | Several years ago, I mentioned offhandedly to a colleague that I had noticed that, if you extend the $\mathsf E\_n$ series downwards in the natural way, by removing nodes from the long arm of $\mathsf E\_8$, then the fundamental group of $\mathsf E\_n$ (i.e., the quotient of the weight lattice by the root lattice) is c... | https://mathoverflow.net/users/2383 | Why is the fundamental group of $\mathsf E_n$ cyclic of order $9 - n$? | If we define the $E\_n$ lattice in the algebraic geometer's way as the orthogonal complement of $(1,\dots, 1, 3)$ in the Lorentzian unimodular lattice $I\_{n,1}$, i.e. in $\mathbb Z^{n+1}$ with intersection form $$\langle(x\_1,\dots, x\_{n+1}), (y\_1,\dots, y\_{n+1})\rangle =x\_1y\_1 + x\_2 y\_2 + \dots + x\_n y\_n - x... | 8 | https://mathoverflow.net/users/18060 | 430743 | 174,461 |
https://mathoverflow.net/questions/430706 | 14 | The well-known Sylvester–Gallai Theorem states that a set of $n>2$ points in $R^2$ not all on a line contains two points such that the line passing through these two points does not contain a third point in the set.
One of the reasons this is somewhat tricky to prove is that it is false in some geometries, including ... | https://mathoverflow.net/users/630 | Sylvester–Gallai theorem for small sets in a finite field | The answer is no, basically thanks to [Menelaus's theorem](https://en.wikipedia.org/wiki/Menelaus%27s_theorem). First observe as a routine application of the [Bombieri-Vinogradov theorem](https://en.wikipedia.org/wiki/Bombieri%E2%80%93Vinogradov_theorem) that there are infinitely many primes $p$ such that $p-1$ contain... | 12 | https://mathoverflow.net/users/766 | 430748 | 174,462 |
https://mathoverflow.net/questions/430744 | 4 | In 1979, Hakimi and Schmeichel [1] initiated such a study by determining the maximum number of triangles and 4-cycles possible in an $n$-vertex planar graph (see also [2] for a small correction).
* [1] S. Hakimi, E. Schmeichel. On the Number of Cycles of Length k in a Maximal Planar Graph. J. Graph Theory 3 (1979): 6... | https://mathoverflow.net/users/171032 | Is there any study on the bounds on the number of even cycles for planar bipartite graphs? | Every $n$-vertex planar graph has at most $O(n^k)$ copies of $C\_{2k}$. Note that the bipartite assumption is not needed. A more general result is proven in my paper [Subgraph densities in a surface](https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/subgraph-densities-in-a-surface/... | 2 | https://mathoverflow.net/users/2233 | 430764 | 174,467 |
https://mathoverflow.net/questions/430749 | 3 | I'm helping in the translation of an article on code theory and I got the following: $t$-frameproof and $t$-wise intersecting, but I don't know what its correct translation into Spanish would be. Can you help me please?
the definition goes like this:
>
> we say that a code $\mathcal{C}$ is $t$-frameproof, or $t$-... | https://mathoverflow.net/users/218991 | What would be the correct Spanish translation? | These articles on encoding [[1](https://upcommons.upc.edu/bitstream/handle/2117/193058/Fernandez-Soriano-Domingo-Seb%C3%A9.pdf?sequence=1&isAllowed=y) and [2](https://web.ua.es/en/recsi2014/documentos/papers/codigos-con-propiedades-de-localizacion-basados-en-matrices-de-bajo-sesgo.pdf)] in Spanish gives some suggestion... | 1 | https://mathoverflow.net/users/11260 | 430765 | 174,468 |
https://mathoverflow.net/questions/430775 | 3 | A graph is [planar](https://en.wikipedia.org/wiki/Planar_graph) if it can be drawn on the plane in such a way that its edges do not cross each other.
A graph is [$k$-planar](https://en.wikipedia.org/wiki/1-planar_graph) if it can be drawn on the plane in such a way that each of its edges is crossed at most $k$ times.... | https://mathoverflow.net/users/158328 | Planar graphs - more or less | These are the graphs with *pairwise crossing number* or *pair-crossing number* at most $k$. Note that it is an open problem whether the pair-crossing number is actually equal to the usual [crossing number](https://en.wikipedia.org/wiki/Crossing_number_(graph_theory)) of a graph. See [Crossing number, pair-crossing numb... | 5 | https://mathoverflow.net/users/2233 | 430777 | 174,470 |
https://mathoverflow.net/questions/430785 | 6 | It is "well-known" (e.g. stated [here](https://math.stackexchange.com/a/56117/27659) without proof and sketched [here](https://math.stackexchange.com/a/123660/27659)) that $\mathrm{Z\_2}$ proves $\mathrm{Con(PA)}$ using the "usual" model-theoretic proof, that is one can build a notion of model $\mathbb{N}$ and satisfac... | https://mathoverflow.net/users/36103 | How does one prove the consistency of $\mathrm{PA}$ in $\mathrm{Z_2}$? | I don't know a detailed reference, but here's a more detailed proof sketch.
The key point is that the "naive" understanding of $\models$ admits a $\Sigma^1\_1$ definition, which $\mathsf{Z\_2}$ can unproblematically implement. (There are various ways to sharpen this of course.)
One way to do this is via **Skolemiza... | 4 | https://mathoverflow.net/users/8133 | 430786 | 174,471 |
https://mathoverflow.net/questions/430759 | 5 | Pythagorean number of a totally real field $\mathbb{K}$ is the minimal number $N$ of
squares $t\_k^2$ required to represent a totally positive $0\leq x\in \mathbb{K}$ as $x=\sum\_{k=1}^N t\_k^2$, where $t\_k\in\mathbb{K}$, $1\leq k\leq N$. E.g. for $\mathbb{K=Q}$ one has $N=4$, by [Lagrange 4-squares theorem](https://e... | https://mathoverflow.net/users/11100 | Pythagorean numbers of real cyclotomic fields | **Content warning**: This answer assumes knowledge of Hasse-Minkowski and mild (local) class field theory. The latter could be eliminated with some more explicit computations in local fields. References to all required facts can be found in Milne's notes or Cassels-Frohlich, as well as Serre's a course in arithmetic f... | 6 | https://mathoverflow.net/users/488927 | 430788 | 174,473 |
https://mathoverflow.net/questions/430771 | 5 | $\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was [asked in MSE](https://math.stackexchange.com/q/4533187/37763).
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and let $M$ be a $G$-module,
that is, an abelian group on which $G$ act... | https://mathoverflow.net/users/4149 | Restriction vs. multiplication by $n$ in Tate cohomology | Let $G = \Bbb Z/4$, acting on the Gaussian integers $M = \Bbb Z[i]$ via multiplication by $i$. The transfer $M\_G \to M^G$ is given by multiplication by $1 + i + (-1) + (-i) = 0$, so
$$H^{-1}(G,M) = M\_G = \Bbb Z[i] / (1-i) \cong \Bbb Z/2.$$
In particular, all elements are 2-torsion.
Similarly, if $H$ is the index-2 ... | 7 | https://mathoverflow.net/users/360 | 430789 | 174,474 |
https://mathoverflow.net/questions/430781 | 1 | Suppose $x\_i$ come from 2-d standard Normal centered at 0. What is the range of $a$ for which the following iteration converges almost surely?
$$w\_{i+1} = w\_i-a x\_i \langle w\_i, x\_i \rangle$$
For 1-d, one can write $w\_{i+1}^2$ as $w\_i^2=w\_0^2\prod\_i (1-\alpha x\_i^2)$, take log and apply central limit the... | https://mathoverflow.net/users/7655 | Range of $a$ such that $w \leftarrow w-a x \langle w, x \rangle$ converges almost surely? | As you say, the threshold is around $a=0.937087$.
The exact condition for convergence is
$$
\int\_{0}^{2\pi}\int\_0^\infty\log(1-2ar^2\cos^2\theta+a^2r^4\cos^2\theta)re^{-r^2/2}\,dr\,d\theta<0.
$$
I don't think there will be a clean formula for $a$.
The derivation of the expression is as follows:
Let $r\_j=\|w\_j\|... | 5 | https://mathoverflow.net/users/11054 | 430790 | 174,475 |
https://mathoverflow.net/questions/430756 | 1 | Let $S$ be a reducible compact complex analytic space, thus we have the decomposition $S=\bigcup\_{i=1}^n {V\_i}$ where $V\_i$ is the irreducible component of $S$. Let $L$ be a line bundle on $S$, I wonder if we can give some descriptions about $H^0(S,L)$ and $H^0(V\_i,L)$.
I have seen this question [Globally generat... | https://mathoverflow.net/users/141609 | Global sections of a line bundle on a reducible complex space | In any case $H^0(S,L)$ is a subspace in $\oplus H^0(V\_i,L)$, but the way it sits there depends on the way the components $V\_i$ are glued together. In the simplest case where they for a simple normal crossing configuration, there is an exact sequence
$$
0 \to H^0(S,L) \to
\bigoplus H^0(V\_i,L) \to
\bigoplus\_{i < j}... | 1 | https://mathoverflow.net/users/4428 | 430792 | 174,476 |
https://mathoverflow.net/questions/430796 | 4 | Let $[n] := \{1,\dots,n\}$, for some large integer $n$, and let $\mathcal{F}$ be a family of 2-element subsets of $[n]$.
The famous [Erdös-Ko-Rado (EKR) theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem) says that if $|\mathcal{F}| > {n - 1 \choose 1} = n-1$, then $\mathcal{F}$ must co... | https://mathoverflow.net/users/491512 | On the number of disjoint subsets of a large set families | Your first question is simply asking what is the minimum number of edges an $n$-vertex graph must have to force a [matching](https://en.wikipedia.org/wiki/Matching_(graph_theory)) of size $m$. This number was determined exactly by a classic result of [Erdős and Gallai](https://citeseerx.ist.psu.edu/viewdoc/summary;jses... | 1 | https://mathoverflow.net/users/2233 | 430808 | 174,478 |
https://mathoverflow.net/questions/430809 | 3 | This question is similar to [a question I asked last year](https://mathoverflow.net/questions/402333), but I'm not asking for the same thing.
---
Let $S$ be a set of ordinals, and consider the Levy collapse $\operatorname{Col}(\omega,S)$. Let $G$ be $\operatorname{Col}(\omega,S)$-generic. Must every unbounded sub... | https://mathoverflow.net/users/146831 | Unbounded subset of $\omega$ in $V[G]$ has an unbounded subset in $V$? | No, this does not hold.
Given an infinite set $A$, let $\hat{A}=\{\langle n, a\_1,...,a\_n\rangle: n\in\mathbb{N}\}$ where $\langle\rangle$ is an appropriate "tupling" operation and $a\_i$ is the $i$th smallest element of $A$. Note that any infinite subset of $\hat{A}$ lets us reconstruct all of $\hat{A}$ itself. Thi... | 5 | https://mathoverflow.net/users/8133 | 430810 | 174,479 |
https://mathoverflow.net/questions/430800 | 9 | For positive integer $n$ the following value of a hypergeometric function
$$\_4F\_3(n,n,n,2n,1+n,1+n,1+n,-1)$$
based on the first few terms looks like
$$ R\_1(n) + R\_2(n) \pi^2$$
where $R\_{1,2}(n)$ are rational numbers. Is there a way to express $R\_{1,2}(n)$ in a closed form using “simple” operations like po... | https://mathoverflow.net/users/41312 | Closed form for ₄F₃(n,n,n,2n;1+n,1+n,1+n;−1) | Fiddling with Maple, I get: if $n$ is a positive real number, then
$$
{{\_4\mathrm F\_3}(n,n,n,2\,n;\,n+1,n+1,n+1;\,-1)}={\frac {{n}^{2}
\sqrt {\pi}\,\Gamma(n+1)\,\psi^{(1)}(n)}{{4}^{n}\,\Gamma \left( n
+{\frac{1}{2}}\right)}. }
$$
Here, $\psi^{(1)}$ is the [trigamma function](https://en.wikipedia.org/wiki/Trigamma_fun... | 13 | https://mathoverflow.net/users/454 | 430811 | 174,480 |
https://mathoverflow.net/questions/430797 | 5 | Let $P \in \mathbb Z[X]$ be monic, separable, of degree $d$, $K$ its splitting field over $\mathbb Q$ and $G$ the Galois group of $K$ over $\mathbb Q$.
Now, let $p$ be a prime number unramified in $K$. If $P \text{ mod } p$ factorizes as a product of $n\_1$ linear factors, $n\_2$ quadratic irreducible factors, $\dots... | https://mathoverflow.net/users/133679 | Cycle type in Galois group from ramified primes | It does *not* imply there is such a permutation in the Galois group, and your question has an interesting history.
First of all, I would say the question you ask is arguably the wrong one: the more natural object to focus on is a prime ideal factorization of $p$, not a factorization of a polynomial mod $p$.
Let $F ... | 10 | https://mathoverflow.net/users/3272 | 430814 | 174,481 |
https://mathoverflow.net/questions/430701 | 1 | Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe under some extra assumption on the relation between the metric and the topology)?
For example, assuming that all closed ... | https://mathoverflow.net/users/121875 | Name of a space with both a topology and a metric that are not compatible? | A "topometric space" is a triple $(X,\tau,\rho)$ where $\tau$ is a topology on $X$ and $\rho$ is a metric on $X$ which is lower semi-continuous with respect to the topology. These are useful in continuous logic. More details can be found in papers by Ben Yaacov.
| 3 | https://mathoverflow.net/users/128723 | 430820 | 174,482 |
https://mathoverflow.net/questions/430825 | 7 | I need the dimensions of the Atkin-Lehner eigenspace for the paper I'm writing.
As is well known, the cuspidal space $S\_{k}(\Gamma\_{0}(N))$ can be decomposed by Atkin Lehner involution. For example, when $N=21=3\times7$, we have the decomposition
$$
S\_{k}(\Gamma\_{0}(N))=S^{(++)} \oplus S^{(+-)} \oplus S^{(-+)} \o... | https://mathoverflow.net/users/491555 | How to get the dimension of Atkin-Lehner eigenspace or do you have any data already obtained? | You can compute these dimensions using *modular symbols* (an auxiliary space which has the same Hecke action as modular forms, but is easier to compute). Here's a Sage example for weight 4 cusp forms of level Gamma0(17):
```
sage: S=ModularSymbols(Gamma0(17), weight=4, sign=1).cuspidal_submodule()
sage: S.atkin_lehn... | 10 | https://mathoverflow.net/users/2481 | 430826 | 174,484 |
https://mathoverflow.net/questions/430836 | 5 | Might there be a research team that has formalised the Riemann Hypothesis? So far I have encountered two related questions:
1. [Is there a formulation of the Riemann Hypothesis in first-order arithmetic?](https://mathoverflow.net/questions/31846/is-the-riemann-hypothesis-equivalent-to-a-pi-1-sentence)
2. [Can the Rie... | https://mathoverflow.net/users/56328 | Formalisation of the Riemann Hypothesis | I just learned of the following formalisation led by Brandon Gomes and Alex Kontorovich from Andrej Bauer(via email):
>
> Brandon Gomes & Alex Kontorovich. Formalization of the Riemann
> Hypothesis in the Lean Theorem Prover. Github repository. 2020.
> <https://github.com/bhgomes/lean-riemann-hypothesis>
>
>
>
... | 8 | https://mathoverflow.net/users/56328 | 430854 | 174,490 |
https://mathoverflow.net/questions/430835 | 1 | $\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion:
$$
\SU(N)\ni m = a\, u \, b
$$
where $a,b$ are independent diagonal $\SU(N)$-matrices each of which accounts for $N-1$ parameters
while $u\in \SU(N)$ is parametrized by the remain $(N-1... | https://mathoverflow.net/users/145660 | On Euler angles decomposition of $\mathrm{SU}(N)$ | Building on the paper [Idel and Wolf - Sinkhorn normal form for unitary matrices](https://arxiv.org/abs/1408.5728) Colin McQuillan [suggested](https://mathoverflow.net/questions/430835/on-euler-angles-decomposition-of-mathrmsun#comment1108718_430835), it is easy to see that every $\operatorname{SU}(N)$ matrix $m$ can b... | 1 | https://mathoverflow.net/users/145660 | 430858 | 174,493 |
https://mathoverflow.net/questions/422450 | 0 | In the post (cross-posted in Mathematics Stack Exchange with identificator MSE [**4244256**](https://math.stackexchange.com/questions/4244256/conjectures-inspired-in-the-context-of-casas-alvero-conjecture-via-the-logarith) and same title) we assume that $P(x)=a\_0+a\_{1}x+\ldots+a\_{n-1}x^{n-1}+a\_{n}x^n$ is a polynomi... | https://mathoverflow.net/users/142929 | Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomial | I'm probably making a stupid mistake, but is Conjecture 1 possibly rather easy? At $\ell=0$ we're assuming
$$ p(x) = \frac{a\_n}{n!} \left( \frac{n}{\frac{d}{dx} \log p(x)} \right)^n $$
Write $\frac{d}{dx} \log p(x) = \dfrac{q(x)}{r(x)}$, a reduced rational expression. Then the right hand side of the hypothesis is
... | 1 | https://mathoverflow.net/users/88133 | 430859 | 174,494 |
https://mathoverflow.net/questions/430868 | 8 | Let $P\in \Bbb{Z}[X]$ be a polynomial with degree $d>1$.
It is conjectured that for all such $P$, their range for integer inputs $R\_P:=P(\Bbb{Z})$ has finite intersection with the set of factorials $\{n!:n\ge 0\}$.
We say that $P$ is “good” if there does not exist some $Q\in \Bbb{Z}[X]\setminus \{X\}$ such that $P... | https://mathoverflow.net/users/130484 | Divisibility chains and polynomials | Every $P$ is a counterexample. Indeed, given a polynomial $P$ consider the recursive sequence $b\_{n+1}=f(b\_n)$ where I take $f(x)=x+P(x)$, say. Then $P(b\_{n+1}) = P(b\_n + P(b\_n)) \equiv P(b\_n) \equiv 0 \bmod P(b\_n)$ since $x-y \mid P(x)-P(y)$ in general. Setting $a\_n=P(b\_n) \in R\_P$ this says that $a\_{n+1}$ ... | 8 | https://mathoverflow.net/users/31469 | 430870 | 174,497 |
https://mathoverflow.net/questions/430866 | 5 | Consider a [set theory](https://en.wikipedia.org/wiki/Alternative_set_theory) with the following axioms:
1. [separation](https://en.wikipedia.org/wiki/Axiom_schema_of_specification): $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$
2. [reflection](https://en.wikipedia... | https://mathoverflow.net/users/74578 | How strong is separation + reflection of unbounded quantifiers? | This theory is mutually interpretable with second-order arithmetic $\mathsf{Z}\_2$ and $\mathsf{ZFC}-\mathsf{PowerSet}$ (and hence equiconsistent with them). Note that the mentioned theories are well-known to be mutually interpretable: the interpretation of $\mathsf{ZFC}-\mathsf{PowerSet}$ in $\mathsf{Z}\_2$ is achieve... | 7 | https://mathoverflow.net/users/36385 | 430883 | 174,501 |
https://mathoverflow.net/questions/430887 | 14 | A [Seifert surface](https://en.wikipedia.org/wiki/Seifert_surface) of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to find the genus of a knot?
Note that it’s been known since the 1980’s that... | https://mathoverflow.net/users/5017 | Is there an algorithm for the genus of a knot? | Jaco and Oertel's paper *[An algorithm to decide if a three-manifold is a Haken manifold](https://www.sciencedirect.com/science/article/pii/0040938384900399)* [1984], plus a bit of work, gives a doubly exponential time algorithm to compute the Seifert genus. (In practice their algorithm lies in exp-poly.)
Agol, Hass,... | 12 | https://mathoverflow.net/users/1650 | 430895 | 174,507 |
https://mathoverflow.net/questions/430906 | 2 | Let $\lambda$ be a limit ordinal and let $T$ be a tree such that for every element $t \in T$ and every $\beta < \lambda$, there is a branch of length at least $\beta$ that contains $t$. Does it follow that $T$ has a branch of length $\lambda$?
| https://mathoverflow.net/users/5199 | Existence of a branch of limit ordinal length | No. Aronszajn trees are the classical example here. The formal definition of an Aronszajn tree is simply a tree of height $\omega\_1$ where every level is countable. This tells you nothing about your requirement. However, the standard construction, indeed the one due to Aronszajn, is the "model tree", in a sense.
Thi... | 4 | https://mathoverflow.net/users/7206 | 430907 | 174,509 |
https://mathoverflow.net/questions/430860 | 4 | Let $\text{Mod}\_g$ be the mapping class group of a closed oriented genus-$g$ surface $\Sigma\_g$ and let $H = H\_1(\Sigma\_g;\mathbb{Q})$. Fix some $r \geq 0$. It is known that the cohomology group $H^k(\text{Mod}\_g;H^{\otimes r})$ is independent of $g$ once $g$ is sufficiently large relative to $g$ and $r$. Does any... | https://mathoverflow.net/users/491583 | Stable cohomology of mapping class group with coefficients in $H^{\otimes n}$ | Appendix B of Randal-Williams' ["Cohomology of automorphism groups of free groups with twisted coefficients"](https://link.springer.com/article/10.1007/s00029-017-0311-0) gives a stable description of the graded $\mathbb{Q}[\Sigma\_q]$-module $H^\*(\Gamma\_g;H^{\otimes q})$.
| 4 | https://mathoverflow.net/users/32022 | 430914 | 174,510 |
https://mathoverflow.net/questions/430632 | 6 | I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, in frequency domain it's a Gaussian spectrum.
I want to explore more on the frequency spectrum estimation of this kind... | https://mathoverflow.net/users/489481 | Harmonic analysis for a beginner | ***Q1.*** Your question 1 does not have a definite answer, this will very much depend on the uniformity of your data and on the noise level. To characterize your spectrum, the method of spectral estimation with a Gaussian process prior seems well suited. The paper [Bayesian Nonparametric Spectral Estimation](https://ar... | 1 | https://mathoverflow.net/users/11260 | 430916 | 174,511 |
https://mathoverflow.net/questions/418235 | 13 | Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^\*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus\_{\text{$k$ odd}} \Omega^k$ and $\Omega^{\text{even}} := \bigoplus\_{\text{$k$ even}} \Omega^k$ be the bundles of odd and even differential forms, r... | https://mathoverflow.net/users/84963 | When are bundles of odd and even differential forms isomorphic? | I will explain that as long as $n>2$ the real vector bundles $\Omega^{even}$ and $\Omega^{odd}$ over $M$ are isomorphic.
If $n>2$ then $dim(\Omega^{even}) = dim(\Omega^{odd}) = 2^{n-1} > n$ and so $\Omega^{even}$ and $\Omega^{odd}$ are isomorphic if and only if they are stably isomorphic. This follows from obstructio... | 3 | https://mathoverflow.net/users/318 | 430918 | 174,512 |
https://mathoverflow.net/questions/430831 | 2 | **Motivation**: I'm trying to understand the proof of Theorem 3.1 in [Antonelli, Saut, and Sparber - Well-Posedness and averaging of NLS with time-periodic dispersion management](https://arxiv.org/abs/1204.4468). Though in the following I'm raising kind of abstract question.
Consider non-linear Schrödinger equation (... | https://mathoverflow.net/users/173418 | What is standard continuity argument for well-posedness? | I understand continuity argument a bit differently than the other post. Suppose you've shown the inequality
$$\|u\|\_{L^\infty(I,X)} \leq C(\|\varphi\|\_{X} + |I|\|u\|\_{L^\infty(I,X)}^3) \tag{1},$$
for some absolute constant $C>0$ and all intervals $I$ with $|I|\leq\delta$.
Since $u(t\_0)=\varphi$ and your solution ... | 3 | https://mathoverflow.net/users/54316 | 430919 | 174,513 |
https://mathoverflow.net/questions/430924 | 2 | I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the following conditions:
1. $K(0,\cdot) = 0$
2. There is a $r>0$ and a sequence $(\lambda\_n,u\_n)\in [0,1]\times \overline... | https://mathoverflow.net/users/173595 | Example of a compact operator that is not uniformly continuous | Your idea almost works.
Let $E = \ell\_1$ be the space of absolutely summable sequences.
Let
$$ K(\lambda, (u\_k)\_{k\in\mathbb{N}}) = (2\sum \lambda^{1/k} |u\_k|, 0, 0, \ldots) $$
Given any bounded set in $E$, its image is (essentially) a bounded set in $\mathbb{R}$, and so is pre-compact.
$K(0,\cdot)$ is obviou... | 3 | https://mathoverflow.net/users/3948 | 430932 | 174,517 |
https://mathoverflow.net/questions/430926 | 1 | Let $G$ be a compact connected Lie group. We denote by $\mathfrak{g}$ the Lie algebra of $G$ and by $\mathfrak{g}^\*$ the dual space of $\mathfrak{g}$. Let $\mathcal{O}\_r: = G\cdot r$ be a generic coadjoint orbit of $G$.
The coadjoint orbit $\mathcal{O}\_r$ endowed with the Kirillov–Kostant–Souriau $\omega$ is a sym... | https://mathoverflow.net/users/172459 | Question about the Kähler structure on generic coadjoint orbits | Put $T = G\_r$. We may, and do, assume that $\beta = r$, and simply describe a $T$-invariant complex structure on $\operatorname T\_{\mathcal O\_r}(r)$.
Instead of having one 1-dimensional subspace of $\mathfrak g$ for every root $\alpha$, we get a $2$-dimensional subspace $\{X\_\alpha + \overline{X\_\alpha} \mathrel... | 2 | https://mathoverflow.net/users/2383 | 430935 | 174,518 |
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