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https://mathoverflow.net/questions/453397 | 2 | A normal modal propositional logic $\Delta$ has the disjunction property if and only if
>
> For any formulas $A\_1,\dotsc,A\_n$, if $\Box A\_1 \vee \dotsb\vee \Box A\_n \in \Delta$ then $A\_k\in \Delta$ for some $k$ with $1\leq k\leq n$.
>
>
>
Let us say that $\Delta$ has the *extended* disjunction property if... | https://mathoverflow.net/users/78564 | An extension of the disjunction property in modal logic | The extended disjunction property is equivalent to plain disjunction property for all normal modal logics $\Delta$.
Assume that $\Delta$ contains
$$A\_0\lor\bigvee\_{i=1}^n\Box A\_i,$$
where $A\_0$ is box-free.
Let $P$ denote the set of propositional variables occurring in the formula. For each $a\colon P\to\{0,1\}... | 2 | https://mathoverflow.net/users/12705 | 453410 | 182,195 |
https://mathoverflow.net/questions/453393 | 4 | The following question is a direct continuation of [this](https://math.stackexchange.com/questions/4758087/if-mathbbcu-v-xn-mathbbcx-y-for-every-n-geq-1-then-already-m) elaborate question; it is mentioned there at the end:
Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in this c... | https://mathoverflow.net/users/72288 | If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$? | The answer to the Question is ``no".
As an example, let $u=xy$, $v=x+y$ be the elementary symmetric functions in $x$ and $y$. It is well-known that $[\mathbb{C}(x,y): \mathbb{C}(u,v)]=2$, so those two fields are not equal.
On the other hand, consider $K\_f:=\mathbb{C}(u,v,f(x))$ for any nonconstant polynomial $f(x)\in ... | 12 | https://mathoverflow.net/users/127660 | 453411 | 182,196 |
https://mathoverflow.net/questions/453453 | 0 | Let $\kappa\_1>0$, $\beta\in [0, 1]$ and $b: \mathbb R\_+ \times \mathbb R^d \to \mathbb R^d$ such that for all $t\ge0$ and $x,y \in \mathbb R^d$ we have $|b(t, 0)| \le \kappa\_1$ and $|b(t, x) - b(t, y)| \le \kappa\_1 ( |x-y| \vee |x-y|^\beta )$ for all $x,y \in \mathbb R^d$. I'm reading section 1.2 in the paper [Dens... | https://mathoverflow.net/users/99469 | An estimate of the integral of the higher order derivative of a bump function | $\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\na}{\nabla}$Note that
\begin{equation}
(\na^n\rho\_\ep)(x)=\ep^{-d-n}(\na^n\rho)(\ep^{-1}x).
\end{equation}
So,
\begin{equation}
\int\_{\R^d}|(\na^n\rho\_\ep)(x-y)|\,dy
=\ep^{-d-n}\int\_{\R^d}|(\na^n\rho)(\ep^{-1}(x-y))|\,dy \\
=\ep^{-n}\int\_{\... | 1 | https://mathoverflow.net/users/36721 | 453455 | 182,208 |
https://mathoverflow.net/questions/453456 | 3 | Suppose I have a 3-manifold obtained via face identifications of a polyhedron (e.g. the Poincaré sphere presented as a dodecahedron with opposite faces glued). Is there a program that exists for easily working with such manifolds? The only thing I really know how to do is to make a triangulation in Regina, but I'd like... | https://mathoverflow.net/users/499323 | Computer program for polyhedral manifolds | The two standard answers to this question are [Snappy](https://snappy.computop.org/) and [Regina](https://regina-normal.github.io/). If you have a famous manifold, then it will likely be in one of the many censuses that come with the programs. If you have a less-than-famous manifold, then find a surgery description of ... | 1 | https://mathoverflow.net/users/1650 | 453457 | 182,209 |
https://mathoverflow.net/questions/453469 | 7 | I'm interested in asymptotics for the sum
$$\sum\_{n\le x}\frac{n}{\text{rad}(n)}.$$
For my research I only need to know whether or not this is $\mathcal{O}(x)$, but I would appreciate more precise asymptotics. Additionally, a reference would be much appreciated. The sum of $\frac{1}{\text{rad}(n)}$ has been discussed ... | https://mathoverflow.net/users/505902 | Asymptotics on sum of n/rad(n) | One has
\begin{align\*}
\sum\_{n \leq x}\frac{n}{\operatorname{rad}n} & = (1+o(1)) \, x \sqrt{\frac{2}{\log x \log \log x}} \exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty), \\
& = x\exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty) \\
& \neq O(x\, (\log x)^A) \ (x\... | 6 | https://mathoverflow.net/users/17218 | 453476 | 182,212 |
https://mathoverflow.net/questions/453477 | 3 | I was thinking of the following problem. Let $f$ be a Taylor expansion and $a\_k$ the associated coefficients,
$$\forall x\in\mathbb{R},~f(x)\triangleq\sum\_{k=0}^\infty a\_kx^k.$$
Let suppose that we have:
$$\forall x\in\mathbb{R},~f(x)>0.$$
Is it possible to find another expansion such that:
$$\forall x\in\... | https://mathoverflow.net/users/510079 | Other expansion for positive Taylor expansion | I believe it is not possible. Here is an argument for this:
*Disclaimer: All inequalities hereafter are meant elementwise*
Let's consider a discrete version of this problem $x\in\{0,1,\dots,N-1\}$. Then we want to find an invertible $N\times N$ matrix $\beta$ (different from identity), such that for every elementwi... | 5 | https://mathoverflow.net/users/165560 | 453485 | 182,216 |
https://mathoverflow.net/questions/453494 | 0 | For any $A\subseteq \mathbb{N}$ we let the (lower) density of $A$ be defined by $$d(A) = \liminf\_{n\to\infty}\frac{|A\cap\{0,\ldots,n\}|}{n+1}.$$
If $\pi:\mathbb{N}\to\mathbb{N}$ is a permutation (bijection), let $\text{ex}(\pi) = \{n\in\mathbb{N}: \pi(n) > n\}$ be the set of [*exceedances*](https://mathoverflow.net/q... | https://mathoverflow.net/users/8628 | Is there a lop-sided permutation $\pi:\mathbb{N}\to\mathbb{N}$? | Let $\pi(n) = n+1$ for all $n$ except that $\pi(2^{k}-1) = 2^{k-1}$. In other words, permute cyclically $(2,3)$, $(4,5,6,7)$, $(8,9,10,\ldots,15)$, and so on. Then $d(\operatorname{pos}(\pi))=1$ whereas $d(\operatorname{neg}(\pi))=0$.
| 8 | https://mathoverflow.net/users/17064 | 453495 | 182,219 |
https://mathoverflow.net/questions/453482 | 5 | Let $E/F$ be a quadratic extension of nonarchimedean local field, and let $G$ be a reductive group over $E$, and $G(E)$ the $E$-points of $G$. (For my question, one may just assume $G=\operatorname{GL}\_n$.) Now $G(E)$ can be viewed in two different way: the $F$ points of the restriction of scalar $G\_1:=\operatorname{... | https://mathoverflow.net/users/32746 | Two different local Langlands parameters for quadratic extension | This came up in a paper of mine not so long ago, and my coauthors and I were surprised that it wasn't made explicit in the standard references, so we wrote it out ourselves:
*Dembélé, Lassina; Loeffler, David; Pacetti, Ariel*, [**Non-paritious Hilbert modular forms**](https://doi.org/10.1007/s00209-019-02229-5), Math... | 5 | https://mathoverflow.net/users/2481 | 453506 | 182,221 |
https://mathoverflow.net/questions/453502 | 5 | Let $f \in L\_1(\mathbb{R})$ be such that $\operatorname{supp} f \subset [0,1]$, and let $K$ be the gaussian kernel $K(t) := \frac{1}{\sigma \sqrt{2 \pi}} \exp(-t^2/2\sigma^2)$, with some small $\sigma < 1/8$.
Is it true that for some universal constant $c > 0$, we have $\|\mathbf{1}\_{[0,1]} \cdot (K \ast f)\|\_1 \g... | https://mathoverflow.net/users/468679 | Does convolution of a compactly supported function with Gaussian need to have fraction of the $L_1$ mass in the original interval? | No by the usual duality nonsense. Let's say $\sigma=1$ (it doesn't matter). The convolution at $x$ is $e^{-x^2/2}$ times the integral of $g(t)=f(t)e^{-t^2/2}$ against $e^{tx}$. If the estimate $\left|\int\_2^3(f\*K)e^{x^2/2}\,dx\right|\le C\int\_0^1|f\*K|e^{x^2/2}\,dx$ were possible, then there would exist an $L^\infty... | 4 | https://mathoverflow.net/users/1131 | 453510 | 182,222 |
https://mathoverflow.net/questions/453069 | 10 | Is there an example of a smooth projective variety $X$ (say, over $\mathbb{C}$) such that the $\mathbb{C}$-algebra $\bigoplus\_{n\geq 0}H^0(X,K\_X^{-n}) $ is not finitely generated?
| https://mathoverflow.net/users/40297 | Can $\bigoplus_{n\geq 0}H^0(X,K_X^{-n}) $ fail to be finitely generated? | Chenyang Xu claims that $X = \mathbb{P}\_S\left(\mathcal{O}\_S \oplus \mathcal{O}\_S(H) \right)$, where $S$ the blow-up of $\mathbb{P}^2$ along $9$ general points and $H$ is any ample Cartier divisor on $S$, provides such an example. See example 3.8 in his paper [K-stability for varieties with a big anti-canonical clas... | 4 | https://mathoverflow.net/users/37214 | 453524 | 182,227 |
https://mathoverflow.net/questions/453514 | 0 | The following question is a direct continuation of [this](https://mathoverflow.net/questions/453393/if-mathbbcux-y-vx-y-fx-mathbbcx-y-for-every-fx-in-mathbb/453411?noredirect=1#comment1173100_453411) question:
Let $u,v \in \mathbb{C}[x,y]$.
Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$... | https://mathoverflow.net/users/72288 | $\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$? | Actually, there are still counterexamples of essentially the same nature as the one given in the previous (linked) thread: Take $u=x+xy$ and $v=x^2y$, so that $\mathbb{C}(u,v)$ is the field of functions symmetric in $x$ and $xy$ (i.e., the fixed field of the involution $\sigma:x\mapsto xy$ and $y = \frac{xy}{x}\mapsto ... | 7 | https://mathoverflow.net/users/127660 | 453528 | 182,229 |
https://mathoverflow.net/questions/453522 | 17 | Is there a characterisation of measurability in a forward way similar to the closure characterisation of continuity?
A function $f\colon X \to Y$ is continuous if equivalently:
1. If $G\subset Y$ is open, then is $f^{-1}(G)$ open too.
2. $f(\overline A) \subset \overline{f( A)}$ where $\overline A$ is the closure o... | https://mathoverflow.net/users/57923 | Forward definition of measurability | If $(X,M)$ is an algebra of sets, then define a relation $\delta\_M\subseteq P(X)^2$ by setting $A\mathrel{\delta\_M}B$ if there does not exist some $S\in M$ with $A\subseteq S$, $B\subseteq S^c$. Suppose that $(X,M)$, $(Y,N)$ are algebras of sets. Let $f:X\rightarrow Y$ be a function. Then it is not too hard to prove ... | 13 | https://mathoverflow.net/users/22277 | 453530 | 182,230 |
https://mathoverflow.net/questions/453531 | 0 | For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa\_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\mathbb R^d)$ be the space of real-valued measurable functions on $\mathbb R^d$. Let $L^0\_+ := L^0\_{+} (\mathbb R^d)$ the subs... | https://mathoverflow.net/users/99469 | Does $c$ in the embedding inequality $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$ depend on $\kappa$? | $\newcommand\ka\kappa$Yes, the best constant $c$ depends on $\ka$ if $p<p'<\infty$.
Indeed, the inequality in question can be rewritten as
$$t^Q\sup\_{\|f\|\_{L^p} \le 1} \|P^\ka\_t f\|\_{L^{p'}} \le c \tag{1}\label{1}$$
if $t>0$ and $1 \le p \le p' \le \infty$, where $Q:=\frac{d(p'-p)}{2pp'}\in[0,\infty]$. So, the b... | 2 | https://mathoverflow.net/users/36721 | 453535 | 182,231 |
https://mathoverflow.net/questions/453222 | 5 | Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V\_K = K \otimes\_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V\_K) \cong K \otimes\_k H^n(G,V)$?
If no, what about with some extra assumptions (for example $G$ is finite and $k$ i... | https://mathoverflow.net/users/510150 | Extension of base field for modules of groups and cohomology | There is a commutative diagram
$\require{AMScd}$
\begin{CD}
k[G]\text{-Mod} @>{-\otimes\_{k[G]}K[G]}>> K[G]\text{-Mod}\\
@V{-^G}VV @V{-^G}VV\\
k\text{-Vect} @>{-\otimes\_kK}>> K\text{-Vect},
\end{CD}
i.e., $(K[G]\otimes\_{k[G]}V)^G\simeq K\otimes\_kV^G$, since $K$ is a free $k$-vector space (and hence $K[G]\simeq K\oti... | 5 | https://mathoverflow.net/users/123673 | 453541 | 182,234 |
https://mathoverflow.net/questions/453542 | 0 | Let $G$ be a graph on $>1$ vertices. Recall that its maximum average degree is defined to be
$$M(G) = \max\left\{\frac{2\left|E\_H\right|}{|V\_H|} \colon H \subseteq G, |V\_H| > 1\right\},$$
and its arboricity is equal to
$$A(G) = \max\left\{\left\lceil\frac{|E\_H|}{|V\_H|-1}\right\rceil \colon H \subseteq G, |V\_H| > ... | https://mathoverflow.net/users/512181 | Arboricity and average degree | Denote $A(G)=A$. Let $H$ be the subgraph for which $\lceil \frac{|E\_H|}{|V\_H|-1}\rceil=A$. Write $e$ for $|E\_H|$, $v$ for $|V\_H|$. We have $e/(v-1)>A-1$, thus $e>(v-1)(A-1)$. On the other hand, $e\leqslant v(v-1)/2$. Therefore $v/2>A-1$, $v>2A-2$, $v\geqslant 2A-1$. Then $$M(G)\geqslant \frac{2e}v=\frac{2e}{v-1}\cd... | 4 | https://mathoverflow.net/users/4312 | 453548 | 182,238 |
https://mathoverflow.net/questions/453539 | 2 | Let $X$ be a curve defined over a number field $K$, and let $G\_K$ be the absolute Galois group of $K$. Let $\text{Aut}(X)$ be the group of $\overline{K}$-defined automorphisms of $X$, and consider the Galois cohomology set $H^1(G\_K, \text{Aut}(X))$.
Denote by $\text{Twist}(X/K)$ the set of twists of $X$ up to $K$-d... | https://mathoverflow.net/users/174655 | Equivalence between twists of a curve and torsors of its automorphism group | Actually, this has little to do with varieties and is just some general phenomenon in topos theory. For simplicity, let $\mathscr C$ be a (small) category with finite limits, endowed with a Grothendieck pretopology. For a sheaf of groups $\mathscr G$ on $\mathscr C$, write $H^1(\mathscr C,\mathscr G)$ for the set of is... | 5 | https://mathoverflow.net/users/82179 | 453551 | 182,239 |
https://mathoverflow.net/questions/453505 | 1 | I'm reading an article regarding the condition for the existence of a solution to a constrained Pick interpolation problem, and the author wrote the following equality - for background, we have the following spaces:
$$H^2\_{\alpha,\beta}\subset M\_k(H^2)$$ where $M\_k(H^2)$ is the set of $k\times k$ matrices with entri... | https://mathoverflow.net/users/489525 | An inner product and projection property in RKHS | I don't fully understand the context here, but it seems to me what you need is the following: Let $A = H^2\_{\alpha, \beta}$, $B = (H^2\_{\alpha, \beta} \cap B\_FM\_k(H^2))^\perp$, then $P\_BP\_A = P\_{A \cap B}$. We can show that this would happen if $A \cap (A \cap B)^\perp \perp B$. (Indeed, we observe that $P\_BP\_... | 1 | https://mathoverflow.net/users/504602 | 453559 | 182,241 |
https://mathoverflow.net/questions/453549 | 0 | If $G =(V,E)$ is a simple, undirected graph (finite or infinite), and $\kappa \neq \emptyset$ is a cardinal, we say that the complete graph $K\_\kappa$ is a **minor** of $G$ if there is a collection ${\frak S}$ of connected and pairwise disjoint subsets of $G$ such that
1. $|{\frak S}| = \kappa$, and
2. whenever $S\n... | https://mathoverflow.net/users/8628 | Hadwiger number of the Hadwiger-Nelson graph on $\mathbb{R}^2$ | No, the set of such cardinals is infinite.
Take $\kappa$ points on the interval $(0, 1) \times \{0\}$. Every two points $(u, 0)$ and $(v, 0)$ have a common neighbour $$\left(\frac{u+v}{2}, \sqrt{1 - \frac{(u-v)^2}{4}}\right)$$ which is not adjacent to any other vertex on the interval. Thus the Hadwiger-Nelson graph h... | 3 | https://mathoverflow.net/users/502833 | 453562 | 182,242 |
https://mathoverflow.net/questions/453442 | 2 | **Definitions and setting**
Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ and let $K\_x:=K(x,\cdot)$.
Let $M(\mathcal{H})$ denote the **multiplier algebra** of $\mathcal{H}$, that is, th... | https://mathoverflow.net/users/36886 | Orthonormal bases in RKHSs via interpolating sequences | It is true that if a sequence $(k\_n)$ is interpolating for $M(\mathcal{H})$, then the *normalized* Kernel vectors $g\_n:=K\_{k\_n}/\Vert K\_{k\_n} \Vert\_\mathcal{H} $ form a Riesz system in $\mathcal{H}$. Of course it does not have to be a complete system, so it is not always a Riesz basis.
To see this, let $g\_n$ ... | 1 | https://mathoverflow.net/users/153260 | 453563 | 182,243 |
https://mathoverflow.net/questions/453556 | 0 | I am interested in $H¹$ right now and the cocycle condition $φ\_{jk} • φ\_{ij} = φ\_{ik}$ because of how it is said to relate to automorphic forms. I can't quite see the relationship between factors of automorphy and $H¹(G,M)$ that is mentioned [here](https://en.wikipedia.org/wiki/Automorphic_form).
Let $G$ be a grou... | https://mathoverflow.net/users/490598 | Modular forms and the cocycle condition in group cohomology | We have $j\_{gh}(x)f(x)=f((gh)x)=f(g(hx))=j\_g(hx)f(hx)=j\_g(hx)j\_h(x)f(x)$, and so $j\_{gh}(x)=j\_g(hx)j\_h(x)$, which is the one-cocycle condition.
| 2 | https://mathoverflow.net/users/460592 | 453564 | 182,244 |
https://mathoverflow.net/questions/453552 | 11 | How does the cardinality of $$\{(a,b): 1 \leq a,b \leq n, \ 2ab/(a+b) \ \mbox{is an integer}\}$$ grow as a function of $n$? What about $$\{(a,b): 1 \leq a,b \leq n, \ \sqrt{ab} \ \mbox{is an integer}\}?$$ For $n=10$, $100$, $1000$, and $10000$ the two functions of $n$ are close and getting closer (in relative terms); o... | https://mathoverflow.net/users/3621 | Asymptotics for pairs of positive integers whose harmonic (resp. geometric) mean is an integer | [CORRECTED 8/28]
I think the asymptotic ratio is $(4/3)\*\ln(2)\approx 0.924196$.
$$G(n) = (6/\pi^2)\*n\*\ln(n) + O(n)$$
$$H(n) = (8\ln(2)/\pi^2)\*n\*\ln(n) + O(n)$$
The proof for the geometric mean goes as Noam Elkies posted.
For the harmonic mean, we will first characterize "primitive" solutions: pairs $(a,b)... | 12 | https://mathoverflow.net/users/2953 | 453565 | 182,245 |
https://mathoverflow.net/questions/453270 | 3 | It is quite well known that for any polyhedral set $K$ and any linear mapping $A$, the set $AK$ is polyhedral and hence closed. I am more curious about the converse problem, namely:
Suppose $K$ is a closed convex cone and for any linear mapping $A$, the set $AK$ is closed. Is $K$ a polyhedral cone?
For closed conve... | https://mathoverflow.net/users/113353 | A converse question about the polyhedrality under linear mapping | I think we can argue as in <https://mathoverflow.net/a/423284/32507> to answer the question in the affirmative.
Let $\mathcal R\_K(x)$ be the radial cone of $K$ at $x$ (as defined in the other answer). Further, for fixed $x \in K$, let $A$ be the projection onto the orthogonal complement of $\operatorname{span}(x)$. ... | 2 | https://mathoverflow.net/users/32507 | 453566 | 182,246 |
https://mathoverflow.net/questions/453574 | 5 | Let A $\subset [0:2]^n$, where $[0:2]=\{0,1,2\}$, then define $2A= \{ a+b\mid a,b \in A \}$. I wanted to know the best known lower-bound estimates for $|2A|$.
I intuitively expect that $|2A| \geq |A|^{\log\_3{5}}$ as this estimate works for $n=1$ and if we take $A= [0:2]^n$, then $|2A| = [0:4]^n$ and the bound fits. ... | https://mathoverflow.net/users/506732 | Estimate of Minkowski sum | This is only a partial answer to your question; I believe there is more current work, and have forwarded your question to someone working in this area to see if they have more recent results.
In Theorem 5 of
*Bourgain, Jean; Dilworth, Stephen; Ford, Kevin; Konyagin, Sergei; Kutzarova, Denka*, [**Explicit constructi... | 9 | https://mathoverflow.net/users/766 | 453587 | 182,255 |
https://mathoverflow.net/questions/453511 | 1 | I am looking for a reference for the following. I believe the set of elliptic curves $E/\mathbb{Q}$ admitting a rational 5-isogeny can be parametrized as
$$E: y^2 = x^3 + f(t)x + g(t), t \in \mathbb{Q}$$
for some polynomials $f,g \in \mathbb{Z}[t]$. The analogous parametrization for elliptic curves admitting a 7-is... | https://mathoverflow.net/users/10898 | Family parametrizing elliptic curves with a rational 5-isogeny | Let $\phi:E\to E'$ be an isogeny of degree $5$, and $j, j'$ be the $j$-invariants of $E$ and $E'$, respectively. Let $r$ be the rational function $$r(z)=\frac{(z^2+10z+5)^3}{z}.$$ Then there is $\tau\in\mathbb Q$ such that $$j=r(\tau)\quad\text{and}\quad j'=r(125/\tau),$$ and the converse holds too.
This result is mo... | 9 | https://mathoverflow.net/users/18739 | 453596 | 182,258 |
https://mathoverflow.net/questions/453595 | 3 | If $\sigma\_k(n)=\sum\_{d\vert n} d^k$, denote
$$F\_1(q)=\sum\_{n\geq1}\sigma\_1(n)\,q^n \qquad \text{and} \qquad
F\_3(q)=\sum\_{n\geq1}n\cdot\sigma\_2(n)\,q^n.$$
>
> **QUESTION.** Assume the prime $p$ is either $2, 3$ or $5$. Is it true that
> $F\_1(q)-F\_3(q)\equiv F\_1(q^p) \pmod p$? Are there other primes for ... | https://mathoverflow.net/users/66131 | Congruences for power-sum of divisors | I wrote an answer in the comment section, so I'll rewrite it here.
---
The claim of
$$F\_1(q)-F\_3(q)\equiv F\_1(q^p) \pmod p$$
is equivalent to the following two cases
$$p\nmid n: \sum\_{d|n} d-nd^3 \equiv 0 \pmod p$$
$$p|n: \sum\_{d|n} d-nd^3 \equiv \sum\_{d| \frac{n}{p}} d \pmod p$$
The second case is ... | 5 | https://mathoverflow.net/users/153549 | 453600 | 182,259 |
https://mathoverflow.net/questions/453608 | 1 | Let us consider a variety $X$ over a field $k$ which is a finite field or an algebraic closure thereof. Let $\ell$ be a prime number different from the characteristic of $k$, and let $\Lambda = \mathbb Z/\ell^k\mathbb Z$ where $k\geq 1$. We have a "derived" category $D\_c^b(X,\Lambda)$ of bounded complexes of construct... | https://mathoverflow.net/users/125617 | Reference for localization distinguished triangles in the derived category of $\ell$-adic sheaves | Indeed $X$ does not have to be smooth.
These triangles are part of the yoga of "recollement". See Section 1.4 of Beilinson-Bernstein-Deligne. A quotable reference that works in great generality is (4.10) of Laszlo-Olsson.
| 4 | https://mathoverflow.net/users/1310 | 453611 | 182,261 |
https://mathoverflow.net/questions/453435 | 2 | Let $\Omega\_1 \subset \mathbb R^2$ be a bounded simply-connected Lipschitz domain, and $f: \bar \Omega\_1 \rightarrow \bar \Omega\_2$ be a homeomorphism, which is a diffeomorphism on $\Omega\_1$ such that $f$ and its inverse have first derivatives in $L^p$ with $p>2$.
The Morrey inequality implies that $f$ is Hölder... | https://mathoverflow.net/users/478527 | Hölderness of the inverse to a $W^{1,p}$-homeomorphism (with additional conditions) of a Lipschitz domain | What you want is not possible. The basic idea is that it is possible for a diffeo $\Omega\_1\to \Omega\_2$ that extends to a homeo $\bar{\Omega}\_1 \to \bar{\Omega}\_2$ to "almost fold up" $\partial\Omega\_1$.
To give an explicit example: let $\Omega\_1$ be the upper half-unit-disk, in polar coordinates the set $\{0<... | 1 | https://mathoverflow.net/users/3948 | 453614 | 182,262 |
https://mathoverflow.net/questions/453609 | 1 | Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini subgroup is cyclic and central. Then $T'$ has order $1$ or $p$ and there are subgroups $X, Y$ such that $T$ = $X\circ Y$ where $X$ is extraspecial and $Y$ has an abelian ... | https://mathoverflow.net/users/488802 | $|C(E):C(E)\cap C(Z(U))|=1$ or $p$ | One possible argument is to compare this with $T/C(Z(U))$ as follows. We have $$C(E)/C(E)\cap C(Z(U))\cong C(E)C(Z(U))/C(Z(U)) \leqslant T/C(Z(U))$$ Now $T/U$ is cyclic, so $T/C(Z(U))$ is cyclic. It acts faithfully on $Z(U)$ by conjugation, and if $x$ is an element of $T$ whose image is a generator of $T/C(Z(U))$ then ... | 4 | https://mathoverflow.net/users/460592 | 453627 | 182,265 |
https://mathoverflow.net/questions/453625 | 5 | For $X\_i$, $i\in[n]$
be a sequence of integrable random variables.
Is there a universal constant $c>0$ such that
$$\mathbb{E}\max\_{i\in[n]}X\_i
\le
c\left(
\max\_{i\in[n]}\mathbb{E}|X\_i|
+
\mathbb{E}\max\_{i\in[n]}|X\_i-X\_{i+1}|
\right),
$$
where $X\_{n+1}:=X\_1$?
This would certainly be true, with $c=1$, if
$
\m... | https://mathoverflow.net/users/12518 | Bounding expected maximum via adjacent differences | I don't think so. Define the $X\_i$'s as follows. Choose $j \in [n]$ uniformly at random, set $X\_j = n$, set $X\_{j+k} = n-|k|\sqrt{n}$ for $1 \le |k| \le \sqrt{n}$ (with indices taken mod $n$), and set $X\_i = 0$ for the remaining $i$. Then $\max\_{i \in [n]} X\_i = n$ almost surely, so $\mathbb{E}\max\_{i \in [n]} X... | 8 | https://mathoverflow.net/users/129185 | 453630 | 182,267 |
https://mathoverflow.net/questions/453632 | 2 | Consider the graph on $\mathbb{Q}\times\mathbb{Q}$ where two members of $\mathbb{Q}\times\mathbb{Q}$ form an edge if and only if their distance is $1$. Is that graph bipartite? If not, what is its chromatic number? (It is [known](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem) to be $\leq 7$.)
| https://mathoverflow.net/users/8628 | Is the Hadwiger-Nelson graph restricted to $\mathbb{Q}\times\mathbb{Q}$ bipartite? | Yes, the chromatic number is $2$. This result is due to Douglas R. Woodall, “Distances realized by sets covering the plane”, *J. Combinatorial Theory* **14** (1973), 187–200. See also Alexander Soifer, *The Mathematical Coloring Book (Mathematics of Coloring and the Colorful Life of Its Creators)*, Springer (2009), §11... | 3 | https://mathoverflow.net/users/17064 | 453633 | 182,268 |
https://mathoverflow.net/questions/453641 | 1 | Let $S$ be a closed real surface having two complex structures $c\_1$ and $c\_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which is holomorphic for both $c\_1$ and $c\_2$. Can we conclude that $\omega=0$ ?
Intuitively I would proceed as follows.... | https://mathoverflow.net/users/481540 | Common holomorphic forms for two distinct complex structures | Any $C^1$ function $g$ on a connected Riemann surface is holomorphic if and only if $dg\wedge \omega=0$, for $\omega$ holomorphic and not the zero form. Where $\omega\ne 0$ this is clear, expanding out in a holomorphic coordinate, say $\omega=f(z)\,dz$, and then integrating along curves to construct a local holomorphic... | 1 | https://mathoverflow.net/users/13268 | 453642 | 182,272 |
https://mathoverflow.net/questions/453640 | 2 | I'm reading the celebrated paper written by Congming Li and Wenxiong Chen, [Classification of solutions of some nonlinear elliptic equations](https://www.researchgate.net/profile/Wenxiong-Chen/publication/38333132_Classification_of_solutions_of_some_nonlinear_elliptic_equations/links/09e415126d86fa7bf2000000/Classifica... | https://mathoverflow.net/users/469129 | What is the infinite Morse index solution? | NEW ANSWER:
Yes there exist infinite Morse index solutions in all dimensions $N\geq 3$. For example you can take the solution in the Li Chen paper
$$
\phi(x,y) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
and trivially cross with $\mathbb{R}$ to define
$$
\phi(x,y,z) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
Clearly this continue... | 4 | https://mathoverflow.net/users/1540 | 453648 | 182,273 |
https://mathoverflow.net/questions/445457 | 1 | Consider a spacetime $(\zeta^{3,1},g)$
where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (related to Dirac/Light cone coordinates/null coordinates). I'm looking to take a [Cauchy foliation](https://ncatlab.org/nlab... | https://mathoverflow.net/users/411249 | Transporting a Cauchy foliation of Minkowski space | We are given a spacetime: $$(\zeta,g)$$ which we recognize as Minkowski space in different coordinates.
We start by observing that the product space of the foliations:
$$\Omega\_{t,r,\theta}(x,y,z)= \varphi\_t(x)\varphi\_r(y)\varphi\_{\theta}(z) \space\space\space\space\space{x,y,z\in(0,1)} \space\space\space{t,r,\... | 0 | https://mathoverflow.net/users/411249 | 453650 | 182,274 |
https://mathoverflow.net/questions/453580 | 6 | This question deals with a concrete exercise from [Geomerty of Schemes](https://www.google.de/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.maths.ed.ac.uk/%7Ev1ranick/papers/eisenbudharris.pdf&ved=2ahUKEwjzsZi6rf-AAxVKxQIHHah3B6oQFnoECBUQAQ&usg=AOvVaw3usdrlkTi1DgXnq_xqASCS) by Eisenbud and Harris but also more... | https://mathoverflow.net/users/108274 | Existence of a reduced fiber implies generically reduced (Exercise III-74 from Geometry of Schemes) | As noted by Jason Starr, the *Generic Principle* has been worked out in details by Grothendieck in EGA IV. If your French is not on top these days, the following version looks quite appealing (and can be found as Theorem 23.9 and 24.4 in Matsumura's *Commutative Ring Theory*) :
Let $f : X \longrightarrow Y$ be a **fl... | 3 | https://mathoverflow.net/users/37214 | 453652 | 182,275 |
https://mathoverflow.net/questions/453655 | 3 | In Montgomery's "[Ten Lectures on the Interface Between Number Theory and Harmonic Analysis](https://doi.org/10.1090/cbms/084)" a bound for the Fourier coefficients of the Selberg polynomial $S^+\_K$ is obtained by using what he calls Vaaler's lemma.
I am interested in obtaining a similar bound directly without using... | https://mathoverflow.net/users/84272 | Fourier coefficients of Selberg polynomials | As it is odd, we can write the Vaaler polynomial $V\_K(x)=\sum\_{1 \le k \le K}c\_{k,K} \sin 2\pi kx$ and its fundamental property is that $|c\_{k,K}| \le \frac{1}{\pi k}$.
This follows easily from its definition in [Vaaler's paper](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-... | 3 | https://mathoverflow.net/users/133811 | 453658 | 182,277 |
https://mathoverflow.net/questions/453659 | 2 | Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E\_S : M \to \mathbb{Z}$ by $E\_S(x) = \left(\text{the number of distinct eigenvalues of } S\_x\right)$. I've seen the following claims in several papers (also see 16.10 in Besse's Einstein Manifolds):
>
> * $M\_S \doteq \left\{x \in M \ \... | https://mathoverflow.net/users/119418 | Why is this subset associated to a $2$-tensor dense? | I claim that the function $E\_S$ is lower semi-continuous, meaning that for any sequence $x\_k \in M$ converging to some $x \in M$, $\lim\_{k \to \infty} E\_S(x\_k) \geq E\_S(x)$.
We want to show that the complement of $M\_S$ has empty interior. Points $x$ in this complement are characterized by the fact that there e... | 4 | https://mathoverflow.net/users/24271 | 453663 | 182,280 |
https://mathoverflow.net/questions/453649 | 9 | Let $x, y$ be finite words over totally ordered alphabet and $<$ denote the lexicographical order, i.e for two not necessarily finite words we say $x < y$ iff one of the following holds
1. There are words $u, x^\prime, y^\prime$ and letters $a < b$ such that $x = uax^\prime, y = uby^\prime$
2. $y = xu$ for some non-e... | https://mathoverflow.net/users/509688 | Elegant proof for $xy < yx \Leftrightarrow x^\mathbb{N} < y^\mathbb{N}$ | Let $x, y \in \Sigma^+$.
Observe that for all $n \in \mathbb{N}$ the inequality $xy<yx$ implies that $$x^ny^n<y^nx^n$$ by repeatedly swapping pairs of $x$ and $y$. It follows that $x^\mathbb{N}\leq y^\mathbb{N}$ (i.e. the two infinite sequences are equal, or $y^\mathbb{N}$ is greater at the first position they differ... | 5 | https://mathoverflow.net/users/502833 | 453676 | 182,284 |
https://mathoverflow.net/questions/428634 | 28 | Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
---
Edited to add: As no answers are forthcoming, does anyone know what happens if we allow arbitrary modules in place of abelian groups?
| https://mathoverflow.net/users/3199 | $A^2$ is isomorphic to $A^{(\omega)}$, but not $A$ | This is not a complete answer, but a construction that might give an answer.
I'll start by constructing a ring with several objects (a.k.a. preadditive
category) $\mathcal{C}$ by generators and relations, using similar ideas as in
Leavitt's ring with $R\not\cong R\oplus R\cong R\oplus R\oplus R$.
The objects $X\_{1... | 4 | https://mathoverflow.net/users/22989 | 453685 | 182,286 |
https://mathoverflow.net/questions/453692 | 4 | $\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL\_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.
It is well known that any $A \in \GL\_n(\mathbb{Z})$ can be written as a product of signed permutation matrices and 'Gauss moves', the latte... | https://mathoverflow.net/users/106337 | Diameter of the unimodular group with Gauss moves | They’re usually called “elementary matrices”, not Gauss moves. Your question is equivalent to asking whether the integer special linear group is boundedly generated by elementary matrices. The answer is no for $n=2$ (an easy exercise using the free product with amalgamation description of the group in that case), but a... | 10 | https://mathoverflow.net/users/317 | 453693 | 182,289 |
https://mathoverflow.net/questions/453678 | 1 | The following question appears in [MSE](https://math.stackexchange.com/questions/4760401/if-langle-f-1-f-2-rangle-langle-g-1-g-2-rangle-then-langle-f-1-lambd) without answers.
Let $f\_1,f\_2,g\_1,g\_2 \in \mathbb{C}[x,y]-\mathbb{C}$.
Assume that $\langle f\_1,f\_2 \rangle = \langle g\_1,g\_2 \rangle \subsetneq \mat... | https://mathoverflow.net/users/72288 | Ideals: If $\langle f_1,f_2 \rangle = \langle g_1,g_2 \rangle$, then $\langle f_1-\lambda,f_2-\mu \rangle = \langle g_1-\delta,g_2-\epsilon \rangle$? | The answer to your questions is no. The ideals $\langle x, y(1-xy) \rangle$ and $\langle x, y \rangle$ are equal, and maximal; but
$$ \langle x-\lambda, y(1-xy)\rangle \neq \langle x-\delta, y-\epsilon \rangle$$
for any $\lambda,\delta,\epsilon \in \mathbb{C}$, $\lambda \neq 0$. (In the notation of your questions, ... | 5 | https://mathoverflow.net/users/88133 | 453703 | 182,291 |
https://mathoverflow.net/questions/453701 | 1 | I have encountered the polynomial equation
$$x^{n+1} = (1 - x)^n ( n + x )$$
where $n \geq 0$, and I am interested in its real roots.
The number $n$ can be an integer or, more generally, any positive real number.
The main reason I am posting this very specific question here is that this could be one of those clas... | https://mathoverflow.net/users/2082 | What can be said about the roots of the polynomial $x^{n+1} - (1 - x)^n ( n + x )$? | The positive root is indeed unique and single for any real $n>0$. Indeed,
1. for $x>1$, we have $$|(1-x)^n(n+x)|=(x-1)^n(n+x)<\left(\frac{n\cdot (x-1)+(n+x)}{n+1}\right)^{n+1}=x^{n+1}$$
by the (weighted) Arithemtic-Geometric Means inequality.
2. for $0\leqslant x\leqslant 1$ there exists a root in $(0,1)$ since the f... | 8 | https://mathoverflow.net/users/4312 | 453705 | 182,292 |
https://mathoverflow.net/questions/453710 | 2 | Let $X$ be an infinite-dimensional Banach space and $C\subseteq X$ be a bounded closed convex subset. Let $\{z\_i\}\_{i\in\mathbb{N}}$ be a sequence of linearly independent points in $C$ and for each $n\in\mathbb{N}$, define $V\_{\leq n} = \operatorname{conv}\{z\_i\}\_{i\leq n}$ and $V\_{\geq n} = \operatorname{conv}\{... | https://mathoverflow.net/users/151332 | Distance between convex hulls in a bounded closed convex set | $\newcommand\la\lambda$The answer is no. E.g., suppose that $X=\ell^\infty$, $z\_1=e\_1$, $z\_2=-e\_1+e\_2$, and $z\_k=e\_2/2+e\_k/k$ for $k\ge3$, where $(e\_1,e\_2,\dots)$ is the standard basis of $\ell^\infty$. Let $C$ be the unit ball in $\ell^\infty$. Then $C$ is a bounded closed convex subset of $X$ and the $z\_k$... | 5 | https://mathoverflow.net/users/36721 | 453724 | 182,298 |
https://mathoverflow.net/questions/445100 | 4 | Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H}^d$ null set. So given $A\_j = f\_j (\mathbb{R}^d)$ with $f\_j$ Lipshitz, one has that A is d-rectifiable if
$$ A \sub... | https://mathoverflow.net/users/109382 | Why is there a $\mathcal{H}^d$-null set in the definition of d-rectifiable set? | If $f:\mathbb{R}^n\to\mathbb{R}^m$, $m<n$ is Lipschitz continuous, then for almost all $x\in\mathbb{R}^m$, the set $f^{-1}(x)$ is ($n-m$)-rectifiable according to the definition that includes the null set $A\_0$. (Such sets are often called countably $(n-m)$-rectifiable). This is a nice generalization of the Sard theor... | 4 | https://mathoverflow.net/users/121665 | 453735 | 182,302 |
https://mathoverflow.net/questions/453688 | 7 | For an abelian group $A$, put $DA=\text{Hom}(A,\mathbb{Z})$ and $D\_{(p)}A=\text{Hom}(A,\mathbb{Z}\_{(p)})$. It is a theorem of Specker that when $A$ is free abelian of countable rank, the natural map $A\to D^2(A)$ is an isomorphism. The proof uses the ordering of $\mathbb{Z}$ in an essential way: at a key step we have... | https://mathoverflow.net/users/10366 | Double dual of free $\mathbb{Z}_{(p)}$-modules | There is at least one proof of Specker's theorem that can be adapted in an obvious way. I believe that the first half of this proof is due to Sąsiada, and the second half to Łoś.
Let $A$ be a free $\mathbb{Z}\_{(p)}$ module of countable rank, and $B=D\_{(p)}A$
its dual, whose elements I will think of as sequences of ... | 9 | https://mathoverflow.net/users/22989 | 453753 | 182,304 |
https://mathoverflow.net/questions/453764 | 4 | Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras.
If $x$ is a positive element of $\mathcal M$ and $y$ is a positive element of $\mathcal N$, it is known that $x\otimes y$ is a positive element of $\mathcal M\overline\otimes\mathcal N$.
So I was wondering : *is every positive element $z$ of $\mathcal M... | https://mathoverflow.net/users/508539 | Approximation from below of positive elements in tensor product of von Neumann algebras | It fails for $M=N=M\_2(\mathbb C)$. Here $M\overline \otimes N = \mathcal B(\mathbb C^2 \otimes \mathbb C^2)$. Let $e\_1 =(1,0), e\_2 = (0,1) \in \mathbb C^2$. The rank 1 projection $z$ onto the span of $\tfrac{1}{\sqrt{2}} ( e\_1 \otimes e\_1 + e\_2 \otimes e\_2)$ gives a counter-example. In fact, let $x\otimes y \leq... | 5 | https://mathoverflow.net/users/126109 | 453774 | 182,310 |
https://mathoverflow.net/questions/453623 | 8 | $\DeclareMathOperator\SL{SL}$As explained in [this question](https://mathoverflow.net/questions/109642/lambda-ring-structures-on-mathbb-a2) , there are two $\lambda$-ring structures on ${\mathbb Z}[x]$. In layman's terms, both come from a realization of ${\mathbb Z}[x]$ as the representation ring of an algebraic (semi)... | https://mathoverflow.net/users/5301 | $\lambda$-ring endomorphisms of ${\mathbb Z}[x]$ | No, it is not cyclic.
For each of the rings, the semigroup of endomorphisms consists of one element of degree $n$ for each positive integer $n$ plus one or two elements of degree $0$. For a positive integer, this is given by the unique product of Frobenius lifts of that degree, i.e. in $R(\mathbb C)$ by $x \mapsto x^... | 4 | https://mathoverflow.net/users/18060 | 453789 | 182,314 |
https://mathoverflow.net/questions/453790 | 1 | Let $\psi\_\alpha(x) = \exp(x^\alpha)-1$ for $\alpha\geq 1$.
Define
$$
\psi\_\infty(x) = \begin{cases}\infty & x>1\\1& x = 1\\ 0 & x <1
\end{cases}
$$
to be such that for any $x>0$ $\psi\_\infty(x) = \lim\_{\alpha\to\infty}\psi\_\alpha(x)$.
Let
$$\lVert X\rVert\_{\psi\_\alpha} = \inf\{k>0\mid \mathbb{E}[\psi\_\alpha(... | https://mathoverflow.net/users/101207 | Is the space of bounded $\psi_\infty$ Orlicz norm random variables equal to $L^\infty$? | For each real $k>0$,
\begin{equation}E\psi\_\infty(|X|/k)=\infty\,P(|X|>k)+P(|X|=k) \\
=\left\{\begin{aligned}\infty\text{ if } P(|X|>k)>0,\\
P(|X|=k)\le1\text{ if } P(|X|>k)=0.
\end{aligned}\right.
\end{equation}
So, indeed, $\|X\|\_{\psi\_\infty}<\infty$ iff $X$ is essentially bounded. Moreover, $\|X\|\_{\psi\_\inft... | 2 | https://mathoverflow.net/users/36721 | 453791 | 182,315 |
https://mathoverflow.net/questions/453787 | 13 | Assume for simplicity that sets $A\_i\subset\mathbb{R}$ are compact. If $A\_1$ and $A\_2$ have positive length, then $A\_1+A\_2$ contains an interval. That is a variant of the classical Steinhaus theorem and it easily follows by looking at neighborhoods of a density point of $A\_1$ and a density point of $A\_2$.
>
... | https://mathoverflow.net/users/121665 | Steinhaus theorem and Hausdorff dimension | The answer to the question is negative. Körner in [Hausdorff dimension of sums of sets with themselves](https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/188/3/90488/hausdorff-dimension-of-sums-of-sets-with-themselves) and Schmeling-Shmerkin in [On the dimension of iterated sumsets](ht... | 14 | https://mathoverflow.net/users/468679 | 453793 | 182,316 |
https://mathoverflow.net/questions/453657 | 0 | $\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian manifold $(M, g)$ that is not [strongly causal](https://en.m.wikipedia.org/wiki/Causality_conditions).
The Lorentzian me... | https://mathoverflow.net/users/503363 | Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds | I would like to argue that the situation considered [in the](https://mathoverflow.net/questions/453657/non-diffeomorphic-but-homeomorphic-under-lorentzian-topology-lorentzian-manifo?noredirect=1#comment1173719_453657) [comments](https://mathoverflow.net/questions/453657/non-diffeomorphic-but-homeomorphic-under-lorentzi... | 5 | https://mathoverflow.net/users/3948 | 453799 | 182,319 |
https://mathoverflow.net/questions/453808 | 3 | Let $X$ be a complex minimal surface of general type, *id est* $K\_X$ is big and nef. It is well-known that $\displaystyle\int\_X3c\_2(X)-c\_1(X)^2\geq0$, and the equality holds if and only if $X$ is uniformized by $\mathbb{B}\subset\mathbb{C}^2$ (the open ball). Ever in this case: $X$ does not contain neither rational... | https://mathoverflow.net/users/57030 | Existence of elliptic curves on surfaces of general type | Take a product $X= C\_1\times C\_2$, where $C\_i$ are smooth curves of genus greater than $1$. $X$ has general type, and is uniformized by a product of two disks. Also $X$ won't contain an elliptic curve because it maps trivially to each $C\_i$.
| 6 | https://mathoverflow.net/users/4144 | 453812 | 182,323 |
https://mathoverflow.net/questions/453815 | 2 | Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set
$$\tag{1}
\int\_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in C\_c^{\infty}\left(\mathbf{R}^n\right)
$$
Then by Theorem 8.38 in Gilbarg & Trudinger's book, in a ball $B\_R$ we shou... | https://mathoverflow.net/users/469129 | Given an eigenvalue equation (elliptic PDE) in a ball $B_R$, prove the convergence of the first nonzero $\lambda_R$ and its eigenfunction $\phi_R$ | For simplicity let me take $H$ smooth or at least $C^\alpha$ so that I don't have to worry about elliptic estimates:
1. Limit as $R\to\infty$:
Use
$$
\lambda\_R = \inf\_{\phi \in C^\infty\_c(B\_R)}\frac{\int |\nabla \phi|^2 + H \phi^2}{\int \phi^2}
$$
Since $C^\infty\_c(B\_R) \subset C^\infty\_c(B\_{R+s})$ for all... | 3 | https://mathoverflow.net/users/1540 | 453820 | 182,324 |
https://mathoverflow.net/questions/453814 | 2 |
>
> Assume that $ A $ is self-adjoint operator and $ B $ is a bounded self-adjoint operator. The definite domain of $ A,B $, denoted by $ D(A) $ and $ D(B) $ satisfies $ D(A)\subset D(B) $. Show that
> \begin{align\*}
> \operatorname{dist}(\sigma(A),\sigma(A+B))\leq \|B\|,
> \end{align\*}
> where $ \sigma(A) $ and $ ... | https://mathoverflow.net/users/241460 | Disturbance of self-adjoint operator | Let $C=A+B$, take a point, say $0$, in the spectrum of $A$ and assume that $[-\|B\|, \|B\|]$, hence $[-\|B\|-\epsilon, \|B\|+\epsilon]$ for some $\epsilon>0$ is contained in the resolvent set of $C$. The spectral theorem, applied to $C$ implies that $\|C^{-1}\| \leq (\|B\|+\epsilon)^{-1}$. Then $A=C-B=(I-BC^{-1})C$ wou... | 2 | https://mathoverflow.net/users/150653 | 453826 | 182,328 |
https://mathoverflow.net/questions/453865 | 9 | Let $G$ be a finite group, let $k$ be a large enough field of characteristic $p>0$. Let $p\mid |G|$.
**Broué's abelian defect group conjecture** states the following:
Let $B$ be a block of $kG$ with
abelian defect group $D$ and let $b\in \text{Bl}(N\_G(D))$ be its Brauer correspondent. Then the derived categories $... | https://mathoverflow.net/users/12826 | Is there a relationship between Broué's abelian defect group conjecture and Alperin's weight conjecture? | Yes. Broué's conjecture implies the abelian defect case of Alperin's weight conjecture. This makes one think there might be a structural statement like Broué's conjecture, for non-abelian defect groups, that would imply Alperin's conjecture in general. Many attempts have been made, to little avail; Rouquier has conject... | 11 | https://mathoverflow.net/users/460592 | 453871 | 182,340 |
https://mathoverflow.net/questions/453881 | 4 | Given a first-order structure $\mathfrak{A}$ and a first-order theory $T$ one can ask if
$$
\varphi(\mathfrak{A}, T) := ``\text{there is a substructure } \mathfrak{B} \text{ of } \mathfrak{A} \text{ such that } \mathfrak{B} \models T \text{''}
$$
holds. I am interested in whether $\varphi$ is absolute between inner... | https://mathoverflow.net/users/29231 | Is the existence of substructures satisfying a theory absolute? | **Assuming $T$ is countable** (in $V$), the answer is yes.
By downward Lowenheim-Skolem applied to $\mathfrak{B}$, $\varphi(\mathfrak{A},T)$ is equivalent to "$\mathfrak{A}$ has a *countable* substructure satisfying $T$." This lets us interpret $\varphi(\mathfrak{A},T)$ as asking about the ill-foundedness of a certai... | 8 | https://mathoverflow.net/users/8133 | 453882 | 182,342 |
https://mathoverflow.net/questions/453837 | 6 | It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat\_\Delta$ and $\mathcal{S}\text{et}\_\Delta$ with the Joyal model structure. This implies that for any fibrant simplicially enriched category, equivalently a topological cat... | https://mathoverflow.net/users/37400 | From the *usual* nerve of topological categories to $\infty$-categories | The answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the singular complex functor with the homotopy coherent nerve functor, and the realization of the singular complex.
Indeed, the ... | 3 | https://mathoverflow.net/users/402 | 453891 | 182,349 |
https://mathoverflow.net/questions/453768 | 5 | Heinrich Martin Weber and David Hilbert created their own class fields in [1891](https://www.sciencedirect.com/science/article/abs/pii/B9780125996617500197) and [1898](https://en.wikipedia.org/wiki/Hilbert_class_field) respectively.
In the past, Weber continued to name $K={Q}(\sqrt{-m}, j(\omega))$, the Kronecker cla... | https://mathoverflow.net/users/167507 | Compare with Weber and Hilbert class field | Keith Conrad discusses the history of class fields in these [lecture notes](https://kconrad.math.uconn.edu/blurbs/gradnumthy/cfthistory.pdf). Weber's and Hilbert's definitions are equivalent, but Weber only considered class fields for ideal groups over $\mathbf{Q}$ or imaginary quadratics.
| 1 | https://mathoverflow.net/users/11260 | 453897 | 182,350 |
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