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https://mathoverflow.net/questions/251324 | 2 | Consider the first-order Hamilton-Jacobi equation (HJ):
$$H(x,u,\nabla u) = 0 \quad \text{ on } \ \Omega,$$ where $\Omega$ is an open set of $\mathbb{R}^n$, $u:\Omega \to \mathbb{R}$, and $H:\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ (*Hamiltonian*) is continuous.
**Definition 1 (Crandall-Lions-Ev... | https://mathoverflow.net/users/nan | Equivalence of alternative definitions of 'viscosity solution' | Only the subsolution case is proven, as the supersolution case is identical.
**Q1**
Suppose $u$ is a subsolution under the definition with strict extremum. Let $\phi$ be a test function such that $u-\phi$ has a **possibly nonstrict** maximum at $x\_0$. Let
$$\psi(x)=\phi(x)+|x-x\_0|^2$$
so that $x\_0$ is a strict m... | 2 | https://mathoverflow.net/users/35874 | 251352 | 114,045 |
https://mathoverflow.net/questions/251361 | 1 | I would like a finite dimensional irreducible representation of ${PSL}\_2(\mathbb{Z})$ which contains a proper sub-representation when restricted to the congruence subgroup $\Gamma(2)$, the kernel of $\mathrm{mod}\_2:{PSL}\_2(\mathbb{Z})\to {PSL}\_2(\mathbb{Z}\_2)$.
This would be used in constructing exact sequences ... | https://mathoverflow.net/users/29625 | Representations of $PSL_2(Z)$ and $\Gamma(2)$. | (Sorry: the earlier statement had to be modified a little; instead, I give a different construction).
Take the standard representation $\rho$ of $PSL(2,\mathbb{Z}$ on $\mathbb{C}^3$ and tensor it with a representation $\theta$ (of dimenstion not one) of the finite group $PSL\_2(\mathbb{Z}/2\mathbb{Z})$ (such a repre... | 3 | https://mathoverflow.net/users/23291 | 251364 | 114,050 |
https://mathoverflow.net/questions/251340 | 1 | I read in some papers the following method:
Let $h$ be a $(1,1)$-tensor field on 3-dimensional Riemannian manifold $(M,g)$ and $p\in M$. Then there exists a smooth local orthonormal basis of the form $\{e\_1,e\_2,e\_3\}$ in a
neighborhood of $p$. Now, let $U\_1$ be the open subset of $M$ where $h\neq 0$ and let $U\_2... | https://mathoverflow.net/users/90655 | Why can we construct the following method? | For question 1, take a smooth function $f(x,y,z)$ positive for $x > 0$ and zero for $x < 0$. Let
$$
h\_0
=
\begin{pmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{pmatrix}
$$
and then let $h=fh\_0$.
Take the standard Euclidean metric on $\mathbb{R}^3$.
Let $e\_1, e\_2, e\_3$ be the standard basis. Clearly $U\_1=(... | 1 | https://mathoverflow.net/users/13268 | 251370 | 114,051 |
https://mathoverflow.net/questions/251371 | 1 | Suppose we have a family $F$ such that:
1. For each $A \in F$ we have $|A| = k$ and $A \subset n$.
2. For each $A,B \in F$ we have $A \cap B \neq \emptyset$.
It is easy to show that there exists a nonempty set $B \subset n$ such that:
1. For each $A \in F$ we have $B \cap A \neq \emptyset$.
2. $|B| \le k^2$.
3. $... | https://mathoverflow.net/users/59012 | Unavoidable finite set for infinite k-intersecting family? | Certainly. Just "pass to the limit". One possible way to do it (in the old fashioned "given $\varepsilon>0$, find $\delta>0$ style"; I will leave recasting it into a slick compactness argument to you) is to consider all finite subsets $F'$ of $F$ and the corresponding $B(F')$. Now choose a set $B$ of largest cardinalit... | 3 | https://mathoverflow.net/users/1131 | 251372 | 114,052 |
https://mathoverflow.net/questions/251357 | 2 | In my probability and numerical analysis research I have come across the following predicament:
>
> If we have a standard normal random variable X with CDF $ \Phi $, and PDF $ \phi $ I am interested in the asymptotic behavior of $\int\_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ where $ a,q \geq 1 $ ar... | https://mathoverflow.net/users/69446 | The asymptotics of $\int_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ for normal distribution using saddle point approximation | In my opinion, that other question has been answered by michael completely in the very first comment. However, since the question arose again, let me just spell the details of michael's answer out.
We have $\Phi(z)=\frac12e^{\sqrt{2/\pi}z+O(z^2)}$. Thus $\Phi(x/a)^{qa}=2^{-qa}e^{\sqrt{2/\pi}qx}e^{O(qx^2/a)}$. Now squ... | 5 | https://mathoverflow.net/users/1131 | 251376 | 114,054 |
https://mathoverflow.net/questions/251354 | 3 | Let $N$ be a prime and let $\mathbb{T} \subset \mathrm{End}(J\_0(N))$ be the Hecke algebra generated over $\mathbb{Z}$ by $U\_N$ and the operators $T\_p$ for primes $p \nmid N$. Fix a maximal ideal $\mathfrak{m} \subset \mathbb{T}$ and let $\mathbb{T}\_{\mathfrak{m}}$ be the completion of $\mathbb{T}$ at $\mathfrak{m}$... | https://mathoverflow.net/users/5498 | Does Gorensteinness of $\mathbb{T}_{\mathfrak{m}}$ imply multiplicity one? | In the ordinary case, the argument is simple so let me recall it here.
The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^{m}$ and the Pontryagin dual $J^{et\*}$ of $J^{et}$ is a free $\mathbb T\_\mathfrak{m}$-module of rank 1 by the ordinar... | 4 | https://mathoverflow.net/users/2284 | 251388 | 114,059 |
https://mathoverflow.net/questions/251395 | 11 | Let $k$ be an arbitrary field. Let $(A, e)$ be an abelian variety over $k$, and let $X$ be a torsor for $A$, i.e. $X$ is a proper smooth $k$-variety, and there is an $A$-action acting $:A \times X \to X$ such that for any $k$-scheme $L$ and a point $x \in X(L)$, the induced "orbit" map $A\_L \to X\_L$ given by $a \maps... | https://mathoverflow.net/users/nan | When $k = \mathbb{F}_q$ finite field, $X$ always has $k$-rational point, and so $A \simeq X$? | This is a theorem of Lang's from 1956. Here's an online document giving a proof (in the form $H^1(A,k)=0$):
Lecture 14: Galois Cohomology of Abelian Varieties over Finite Fields,
William Stein. <http://wstein.org/edu/2010/582e/lectures/582e-2010-02-12/582e-2010-02-12.pdf>
Stein notes that there is a "more modern p... | 23 | https://mathoverflow.net/users/11926 | 251399 | 114,062 |
https://mathoverflow.net/questions/251326 | 3 | Assume $G$ is the Klein four group $G=\{1,\sigma\_1,\sigma\_2,\sigma\_3\}$.
Let $G$ act on $X=\mathbb{A}^2\times\mathbb{P}^1$ via:
$$\sigma\_1\cdot(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu]) \text{ and } \sigma\_2 \cdot ((x,y,[\lambda:\mu])=(x,-y,[\mu:\lambda]).$$
Is the quotient $X/G$ some well known variety? Wha... | https://mathoverflow.net/users/70593 | Is this quotient of a threefold known? What are its singularities? | I denote by $\mathbb{A}^1\_x$ the affine line given by $y=0$ in $\mathbb{A}\_2$ and by $\mathbb{A}^1\_y$ the line $x=0$ in $\mathbb{A}\_2$.
Then, the fixed locus of $\langle \sigma\_1 \rangle$ is $\mathbb{A}^1\_y \times [1:0] \cup \mathbb{A}^1\_y \times [0:1]$.
The fixed locus of $\langle \sigma\_2 \rangle$ is $\m... | 1 | https://mathoverflow.net/users/37214 | 251410 | 114,069 |
https://mathoverflow.net/questions/251415 | 5 | A unit speed geodesic $\gamma:I\to M$ on a Riemannian manifold $M$ can be lifted to a curve $\sigma=(\gamma,\gamma')$ on $SM$, the unit sphere bundle (the unit tangent bundle) of $M$.
Let us say that the geodesic $\gamma$ is dense on the tangent bundle if the trace $\sigma(I)$ of $\sigma$ is dense on $SM$.
The existenc... | https://mathoverflow.net/users/55893 | Simply connected manifolds with dense geodesics on the tangent bundle | Burns and Donnay proved that every surface (including a sphere) admits a Riemannian metric that makes the geodesic flow ergodic with respect to Liouville measure, and hence topologically transitive (there is some $v\in SM$ whose orbit under the geodesic flow is dense in $SM$, in other words, the corresponding geodesic ... | 8 | https://mathoverflow.net/users/5701 | 251420 | 114,073 |
https://mathoverflow.net/questions/251422 | 5 | In Dusart papers he proves that $\prod \limits\_{p \leq x} \frac{p\_i}{p\_i-1} \leq e^\gamma \ln (x) \left(1+\frac{0.2}{\ln ^2 (x)} \right)$ for large numbers.
What I am asking is could we make the bound a little bit sharper by making it like this
$$ \prod \limits\_{p \leq x} \frac{p\_i}{p\_i-1} \leq e^\gamma \ln (x... | https://mathoverflow.net/users/95470 | Even sharper upper bound for prime product? | Yes, although writing down explicitly what "sufficiently large" means might be a challenge.
The prime number theorem implies that there exists a constant $c>0$ such that
$$
\prod\_{p\le x} \frac p{p-1} = e^\gamma \ln (x) \big( 1 + O\big( \exp(-c\sqrt{\ln x}) \big) \big).
$$
From this it is easy to deduce, for every $... | 12 | https://mathoverflow.net/users/5091 | 251424 | 114,074 |
https://mathoverflow.net/questions/251383 | 1 | Let $p$ be a rational prime. Consider the topological group $\prod\_{\ell\ne p} {\mathbb Z}\_{\ell} $. Does there exist a natural number $M$ such that $\{p^{n\_1}+p^{n\_2}+\cdots+p^{n\_m}: \text{$0\le m\le M$ and $n\_i\ge 0$
for each $i$}\}$ is a dense subset of $\prod\_{\ell\ne p} {\mathbb Z}\_{\ell}$?
How about the... | https://mathoverflow.net/users/4948 | Dense subset in a product of $p$-adic groups? | Felipe's idea can be converted into a rigorous proof that there is no such $M$.
Fix a positive $\epsilon < 1/M$. Suppose $m$ is a large positive integer prime to $p$ with the property that the units group $U(\mathbb{Z}/m\mathbb{Z})$ has an exponent $\lambda$ (say) smaller than $m^{\epsilon}$. Then, modulo $m$, the n... | 2 | https://mathoverflow.net/users/16510 | 251426 | 114,075 |
https://mathoverflow.net/questions/251331 | 3 | Consider the following Hamilton-Jacobi (HJ) equation:
$$u\_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ (*Hamiltonian*) is continuous.
Consider the initial condition $$u = g \quad \text... | https://mathoverflow.net/users/nan | Uniqueness of viscosity solutions of Hamilton-Jacobi equation | I would recommend the book by Bardi and Capuzzo-Dolcetta:
"Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations" Bardi, Martino, Capuzzo-Dolcetta, Italo
Read the proofs for stationary equations $H(\nabla u, x) = 0$ on bounded domains first to get the main ideas. The proof in Evans has more b... | 3 | https://mathoverflow.net/users/18406 | 251428 | 114,076 |
https://mathoverflow.net/questions/251397 | 2 | Let R be a local noetherian ring and let M, N be two finitely generated modules.
>
> Is it true that if $M \otimes\_R N$ has finite length, then $Tor\_i^R(M,N)$ also has finite length for all i?
>
>
>
I know a reference for a special case of this in Serre's book on local algebra. But I really don't want to ass... | https://mathoverflow.net/users/16857 | If tensor product has finite length, do higher Tors also have finite length? | To expand on Mohan's comment:
1) If $S$ is a local noetherian ring and $A$ an fg $S$-module, then $A$ has finite length if and only if it is supported on the maximal ideal.
2) If $S$ is a local noetherian ring with residue field $k$ and $A,B$ are fg $S$-modules such that $A\otimes\_SB=0$, then
$$(A\otimes\_Sk)\otim... | 5 | https://mathoverflow.net/users/10503 | 251435 | 114,078 |
https://mathoverflow.net/questions/251425 | 2 | Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK\_X) \otimes H^0(X,nK\_X) \to H^0(X,(m+n)K\_X)$ for $m, n \in \mathbb{Z}\_{>0}$ is surjective?
My case is particularly simple: $X$ is a smooth **curve** of genus greater than $1$, and I wish $\otimes\_{i=... | https://mathoverflow.net/users/29730 | When is the map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ surjective? | I assume you mean $H^0(X, K\_X)^{\otimes m}$ rather than $\oplus\_{i=1}^m H^0(X, K\_X)$. If $X$ is a smooth projective connected complex curve of genus $g \geq 2$, then the map
$$H^0(X, K\_X)^{\otimes m} \longrightarrow H^0(X, m K\_X),$$
is surjective for any $m \geq 0$, as long as $X$ is not hyperelliptic. This is a t... | 15 | https://mathoverflow.net/users/21724 | 251438 | 114,080 |
https://mathoverflow.net/questions/251419 | 10 | Are there $n\times n$ real matrices $A\_{1}, \ldots, A\_{n}$ such that the $n$-homogeneous polynomial
$$
f(x\_{1}, \ldots, x\_{n}) = \det(x\_{1} A\_{1}+\cdots +x\_{n} A\_{n})
$$
never vanishes on $\mathbb{R}^{n}\setminus\{0\}$?
I was listening to a seminar of a student, and a certain problem boils down to this line... | https://mathoverflow.net/users/50901 | Homogeneous polynomials, mixed determinants, positive definiteness | I doubt about powers of 2, it looks that the answer is $n=1,2,4,8$.
Without loss of generality $A\_1=I$ (else replace $A\_i$ to $A\_iA\_1^{-1}$ for all $i$). Then for any $x\in \mathbb{S}^{n-1}$ the vectors $x=A\_1 x,A\_2x,\dots,A\_nx$ should be linearly independent (else $x$ belongs to a kernel of a certain linear ... | 12 | https://mathoverflow.net/users/4312 | 251443 | 114,081 |
https://mathoverflow.net/questions/250957 | 6 | Is there a simple reason why uncountable compact sets of real numbers have cardinality continuum?
I know that this is immediate from the Cantor-Bendixon Theorem, but I wonder whether this consequence in its own can be proved in a simpler/shorter manner.
Perhaps building a Cantor set inside the compact set is one ans... | https://mathoverflow.net/users/2415 | A simpler proof that compact sets have cardinality continuum? | I think "the" conceptual reason why an uncountable compact subset of $K\subset \mathbb{R}$ has the cardinality of the continuum is that it has a "canonical" map onto a space homeomorphic to $[0,1]$.
The complement of $K$ is a union of open intervals: a left ray, a right ray and at most countably many bounded ones.
Th... | 4 | https://mathoverflow.net/users/89334 | 251448 | 114,083 |
https://mathoverflow.net/questions/251458 | 4 | Does anybody know who that first introduced the notion of Killing vector field?
Thanks.
| https://mathoverflow.net/users/90655 | Who first introduced the notion of Killing vector field? | Well, this is a point of contention depending on what precisely you call a Killing vector field.
You can argue that implicitly in the work of Sophus Lie on his namesake groups and algebras the idea of infinitesimal symmetries, and hence the Killing vector fields associated to the bi-invariant metric, are already pre... | 10 | https://mathoverflow.net/users/3948 | 251460 | 114,088 |
https://mathoverflow.net/questions/251339 | 2 | There is a well-known relation between the spectrum of graph laplacian and its complement's laplacian, namely
$$λ\_j (G^c) + λ\_{n+2−j} (G) = n\;,$$
where the eigenvalues $λ\_j$ are sorted in increasing order ($λ\_1$ is always 0), $G^c$ is the graph's complement, and $n$ is the number of vertices. This can be easil... | https://mathoverflow.net/users/40345 | Laplacian spectrum of directed network (digraph) and its complement | This is true even if you allow your digraph to have edges with arbitrary weight. Let's denote the characteristic polynomial of a weighted digraph, $G$, by $P\_{G}(\lambda)=\det\left(\lambda I-L(G)\right)$. The complement $G'$ has characteristic polynomial
$$P\_{G'}(\lambda)=\det\left(\lambda I-L(G')\right)=(-1)^n\det\... | 1 | https://mathoverflow.net/users/2384 | 251465 | 114,089 |
https://mathoverflow.net/questions/251434 | 36 | Suppose we have a finitely presented group $G$ with decidable word problem. Is it decidable to check whether a given element $x\in G$ has finite order or infinite?
| https://mathoverflow.net/users/10443 | Is it decidable to check if an element has finite order or not? | A finitely presented group with decidable word problem and undecidable order problem is in [McCool, James
Unsolvable problems in groups with solvable word problem.
Canad. J. Math. 22 1970 836–838](http://cms.math.ca/10.4153/CJM-1970-094-6).
| 38 | https://mathoverflow.net/users/15934 | 251471 | 114,092 |
https://mathoverflow.net/questions/251472 | 1 | Let $(M,g)$ be a connected Riemannian manifold of dimension $n>1$. Then the Hopf-Rinow theorem states that $(M,g)$ is geodesically complete if and only if $(M,d\_g)$ is complete as a metric space ($d\_g$ is the induced intrinsic metric).
I need to know if a similar result is true under weaker hypothesis:
1) Suppose... | https://mathoverflow.net/users/nan | Generalizations of Hopf-Rinow theorem | It seems that the answer to 1 is no: [The Hopf-Rinow Theorem is false in infinite Dimensions](http://blms.oxfordjournals.org/content/7/3/261.full.pdf). If you add the assumption that $M$ is a locally compact length space then the answer is "yes" by Theorem 2.5.28 in [this book](http://www.math.psu.edu/petrunin/papers/a... | 0 | https://mathoverflow.net/users/4362 | 251477 | 114,095 |
https://mathoverflow.net/questions/251470 | 120 | I was very happy to learn that the work which led to the award of the [2016 Nobel Prize in Physics](https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/) (shared between David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz) uses Topology. In particular, the prize was awarded *"for theoretical di... | https://mathoverflow.net/users/8103 | Topology and the 2016 Nobel Prize in Physics | Roughly speaking: When a system consists of particles interacting strongly, you won't have large movements for any particle unless all of them move together. There is still "quantization", so you'll have jumps in the type of movement (example: the quantum Hall effect). So the topology of the system plays a role. For ex... | 56 | https://mathoverflow.net/users/12310 | 251486 | 114,101 |
https://mathoverflow.net/questions/251246 | 3 | Let $A=(a\_{ij})\_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers. For every positive integer $n$, we shall denote with $A\_n$ the $n \times n$ matrix $A\_n=(a\_{i,j})\_{i,j=1}^{n}$, and if $x \in \mathbb{C}^{n}$, we shall write $||x|| = \sqrt{\sum\_{i=1}^{n} |x\_i|^2}$. In the following, each $x \in \mathbb... | https://mathoverflow.net/users/99197 | Matrices Representing Bounded Operators and Absolute Values | By following Yemon Choi's hint, I give here a complete answer to my question.
We denote by $\mathbb{Z}$ the set of all integer numbers.
Let $f:[-\pi, \pi) \rightarrow \mathbb{R}$ be defined by $f(x)=x$. The Fourier coefficients of $f$ are:
\begin{equation}
c\_{0}=\frac{1}{2\pi} \int\_{-\pi}^{\pi} x dx = 0,
\end{equat... | 1 | https://mathoverflow.net/users/99197 | 251489 | 114,104 |
https://mathoverflow.net/questions/248624 | 23 | The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
*"All told, this suggests that in Top we have been studying
the wrong mathematical objects.The right ones are the spaces in CGHaus."*
CGHaus is the category of compactly generated Hausdorff spaces.
It i... | https://mathoverflow.net/users/6129 | The "right" topological spaces | The convenient category CGH of compactly generated Hausdorff spaces has some poor colimits, since Hausdorffification may change the underlying point sets. The category CGWH of compactly generated *weak* Hausdorff spaces is even better behaved. The advantages are discussed in Chris McCord's paper "Classifying Spaces and... | 13 | https://mathoverflow.net/users/9684 | 251490 | 114,105 |
https://mathoverflow.net/questions/251473 | 3 | I am looking for a proof of the following fact:
If $G$ is a finite subgroup of $SL\_n(\mathbb{C})$ acting on $\mathbb{A}\_{\mathbb{C}}^n$, then the resulting quotient scheme is Gorenstein.
Thanks.
| https://mathoverflow.net/users/48616 | Quotient of affine space by finite subgroup of SL(V) is Gorenstein | You probably want to show that the quotient scheme $\mathbb{A}^n\_{\mathbb{C}}/G$ is Gorenstein.
A proof can be made along the following line. Since $G \subset SL\_n(\mathbb{C})$, the canonical bundle of $\mathbb{A}^n\_{\mathbb{C}}$ is $G$-equivariant and satisfy the following : for all $x \in \mathbb{A}^n\_{\mathbb{... | 5 | https://mathoverflow.net/users/37214 | 251492 | 114,106 |
https://mathoverflow.net/questions/251407 | 3 | This is a follow-up to [this question](https://mathoverflow.net/questions/251283/are-morphisms-of-parametrized-spectra-themselves-parametrized-morphisms-of-spect), in which Denis Nardin nicely explained that
$$
\operatorname{Map}\_{\operatorname{Fun}(X, \operatorname{Sp})}(E\_X, E'\_X)
\simeq
\operatorname{Map}(X, \ope... | https://mathoverflow.net/users/39713 | Morphisms of parametrized ring spectra | So the answer is a bit surprising (maybe I have a mistake). You have an adjunction between $Fun(X,\mathrm{Sp})$ and $\mathrm{Sp}$ which in one direction sends a functor to its (homotopy) colimit and on the other hand sends a spectrum to the constant functor $X \to \mathrm{Sp}$ with that value. If $X$ has an $E\_\infty$... | 2 | https://mathoverflow.net/users/51164 | 251496 | 114,108 |
https://mathoverflow.net/questions/251497 | 0 | Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi\_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a sequence $c\_n>0$ such that
$$
\sum\_{n=1}^\infty c\_n \xi\_n \quad \text{converges almost surely.}
$$
Can one give an ... | https://mathoverflow.net/users/37987 | Taking away the "almost sure" | Let $\Omega=(0,1)$ and $\xi\_n:\omega\in\Omega\mapsto$ the $n$-th term of the continued fraction expansion of $\omega$. Given a sequence $c\_n$, there is another sequence $m\_n\in\mathbb{N}$ such that $\sum\_{n=1}^\infty c\_nm\_n$ diverges. Let $x=[m\_1,m\_2,\dots,m\_n,\dots]$. Then $\sum\_{n=1}^\infty c\_n\xi\_n(x)$ d... | 5 | https://mathoverflow.net/users/37103 | 251498 | 114,109 |
https://mathoverflow.net/questions/251351 | 2 | **Background**: Let $W$ be a finite reflection group of rank $n$, acting on $\mathbb{R}^n$. The reflecting hyperplanes of $W$ meet the unit sphere $S^{n-1}\subset\mathbb{R}^n$, inducing a simplicial complex structure on $S^{n-1}$ ("Coxeter complex of $W$"). $W$ acts freely and transitively on the facets of this triangu... | https://mathoverflow.net/users/12419 | Relation between Riemannian and Cayley-graph distance in a finite Coxeter group | Your suggestion that the Riemannian metric on an orbit and the word metric will be almost homothetic in some sense is very far from the reality.
We consider two metrics on the symmetric group $S\_n$, the word metric $|\cdot|\_w$ and the metric coming from the restriction of the Riemannian metric on the sphere to an o... | 1 | https://mathoverflow.net/users/89334 | 251507 | 114,112 |
https://mathoverflow.net/questions/251499 | 2 | It's probably a very simple question but I am not sure about the reference.
In the definition of a balanced monoidal category we require that the braiding isomorphims $$c\_{V, W}: V \otimes W \to W \otimes V$$ are not arbitrary but admit twisting isomorphisms $\theta\_V: V \to V$ which, in a way, represent $c\_{V, W}... | https://mathoverflow.net/users/11051 | Balanced monoidal and homotopy symmetric | No, a balanced monoidal structure is something different. One way to think about these things is in terms of the relevant operads. Braided monoidal means an algebra over the $E\_2$ operad, while symmetric monoidal means an algebra over the $E\_{\infty}$ operad.
Balanced monoidal means an algebra over the [*framed* $... | 5 | https://mathoverflow.net/users/290 | 251508 | 114,113 |
https://mathoverflow.net/questions/251517 | 3 | Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d\_A: \Omega^k(\Sigma, E) \rightarrow \Omega^{k+1}(\Sigma, E)$ along with a Higgs field $\Phi \in \Omega^{(1,0)}(\Sigma, {\rm End}(E))$. The equ... | https://mathoverflow.net/users/56095 | Transformation between two conventions of Hitchin equation | Let me start with the second point of view: you start with a holomorphic structure $\bar\partial^E$ and a (holomorphic) Higgs field $\Phi.$ If this pair is stable (e.g., as defined in Hitchin's original reference), then there exist a unitary metric $h$ on the bundle such that the Chern connection $\nabla$ of $\bar\part... | 2 | https://mathoverflow.net/users/4572 | 251520 | 114,116 |
https://mathoverflow.net/questions/251430 | 3 | Suppose $M$ is a compact four manifold and $P$ is an $SU(2)$ bundle, let $\mathfrak{g}$ be the adjoint bundle of $P$, given a connection $A$ on this bundle. Given $\phi\in \Omega^1(\mathbb{g})$, we have the following Weitzenbock formula:
$$d\_A d^{\star}\_A\phi+d^{\star}\_Ad\_A\phi=\nabla^{\star}\_A\nabla\_A\phi+\sta... | https://mathoverflow.net/users/25054 | About the Weitzenböck Formula for $SL(2,\mathbb{C})$ connection | I think a get a complete answer for this question.
As $SL(2,\mathbb{C})$ bundle P' can reduce to an $SU(2)$ bundle denote as P and $sl(2,\mathbb{C})$ is $su(2)\oplus isu(2)$, under this decomposition, our connection $\mathbb{A}$ can be write as $\mathbb{A}=A+iB$, here $A$ is an $SU(2)$ connection and $B\in\Omega^1(\... | 1 | https://mathoverflow.net/users/25054 | 251522 | 114,117 |
https://mathoverflow.net/questions/251503 | 8 | In $R^n$ (the real space) we have an open connected set $D$, such that $\partial D$ is triangulable. Can we prove the closure $\bar{D}$ is triangulable or any counterexample?
Furthermore, the $\partial D$ are piecewise algebraic in the question I am considering, I do not know whether this would be helpful for the abo... | https://mathoverflow.net/users/91620 | Boundary triangulation induces triangulation | In many categories, the answer is known to be yes, [see](https://arxiv.org/pdf/math/0403055.pdf)
*Emil Saucan*, MR 2184196 [**Note on a theorem of Munkres**](http://dx.doi.org/10.1007/s00009-005-0040-z), *Mediterr. J. Math.* **2** (2005), no. 2, 215--229.
| 4 | https://mathoverflow.net/users/11142 | 251524 | 114,118 |
https://mathoverflow.net/questions/251526 | 0 | Suppose I am given a countable family $(\mu\_n)$ of finite Borel-measures on a compact interval $[0,T]$. Can I find a dominating measure $\mu$ (with $\mu\_n \ll \mu$ for all $n$), such that all Radon-Nikodym densities $\frac{d\mu\_n}{d\mu}$ are essentially bounded by a constant $K$ (independent of n)?
| https://mathoverflow.net/users/76239 | Dominating measure with bounded Radon-Nikodym density | No, take any finite measure $\mu$ and look at the family $n \mu$ for n integral.
| 4 | https://mathoverflow.net/users/nan | 251534 | 114,123 |
https://mathoverflow.net/questions/251184 | 7 | Is there a separative forcing notion $\mathbb{P}$ such that:
1) For any $p \in\mathbb{P}, \mathbb{P}/p = \{q \in \mathbb{P}: q \leq p \}$ is not forcing isomorphic to any homogeneous forcing notion,
2) For all $G$, $\mathbb{P}$-generic over $V$, $HOD^{V[G]} \subseteq V$.
Here by homogeneity I mean cone homogeneit... | https://mathoverflow.net/users/11115 | Non-homogeneous forcing and HOD | It is consistent that the answer is positive and it is consistent that the answer is negative.
**Claim:** There is a generic extension, $V[G]$ by a weakly homogeneous forcing notion in which there is a rigid forcing notion $\mathbb{P}$ such that for every generic filter $H \subseteq \mathbb{P}$, $$HOD^V = HOD^{V[G]}... | 8 | https://mathoverflow.net/users/41953 | 251539 | 114,126 |
https://mathoverflow.net/questions/251519 | 4 | I am trying to understand the action of a modular S transformation on a $\vartheta$-function. To do this for the problem I'm considering I first need to understand the following.
Given a $\vartheta$-function,
$$
\vartheta\Big[\genfrac{}{}{0pt}{}{\frac{p}{q}}{0} \Big](0|q\,\tau) = \sum\_{n\in \mathbb{Z}} e^{i \pi (n... | https://mathoverflow.net/users/99341 | Modular S transformation on higher order $\vartheta$-functions using Poisson summation | I like the treatment of theta functions in Henryk Iwaniec's book "Topics in Classical Automorphic Forms" - Chapter 10. I'm convinced that Proposition 10.4 of this book proves exactly what you need, although it is a good bit more general than you need, since Iwaniec proves a modular $S$ tranformation for the theta serie... | 1 | https://mathoverflow.net/users/48142 | 251542 | 114,128 |
https://mathoverflow.net/questions/251538 | -1 | I am trying to prove or disprove the next statement that seems necessary for the proof of Proposition 2.9 of this [book](https://books.google.de/books?id=e8tXjbBGEroC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false).
>
> Let $U\subset R^k$ be compact and $f:R^n\times U \to R^m$ be twice differ... | https://mathoverflow.net/users/98759 | Proof of $\lim_{i\to\infty}\lambda_i^{-1}\left|f(\hat{x}+\lambda_ix,u_i) - f(\hat{x},u_i) - D_xf(\hat{x},u_i)(\lambda_ix)\right| = 0$ | By Taylor's theorem with remainder,
$$ | f(\hat{x} + \lambda\_i x, u\_i - f(\hat{x},u\_i) - D\_x f(\hat{x},u\_i)(\lambda\_i x) | \leq C \lambda\_i^2 |x|^2 \sup\_{y\in B\_{\lambda\_i |x|} (\hat{x})} |D\_x^2 f(y,u\_i)| $$
for some universal constant $C$. For sufficiently large $i$, the right hand side is bounded by ... | 1 | https://mathoverflow.net/users/3948 | 251544 | 114,129 |
https://mathoverflow.net/questions/251541 | 3 | We work in $\sf ZFC+GCH$. Let $D$ be a class of uncountable regular cardinals, and for every $\alpha$ let $\Bbb Q\_\alpha$ be either trivial if $\alpha\notin D$, or forcing with these two properties:
1. $|\Bbb Q\_\alpha|=\alpha$, and
2. $\Bbb Q\_\alpha$ is $\alpha$-distributive (any less than $\alpha$ dense open sets... | https://mathoverflow.net/users/7206 | Preserving distributivity with finite support products | GCH implies that $\mathbb{Q}\_\alpha$ is $\alpha$-distributive in the generic extension by $\mathbb{P}\_\alpha$.
Note that $\Vdash\_{\mathbb{Q}\_\alpha} ``\check{\mathbb{P}}\_\alpha$ is $\check\alpha$.c.c.$"$. For successor cardinal $\alpha$ this is clear, as $|\mathbb{P}\_\alpha| < \alpha$. For strongly inaccessibl... | 5 | https://mathoverflow.net/users/41953 | 251551 | 114,131 |
https://mathoverflow.net/questions/251546 | 2 | Let $f\in L^1(\mathbb{R}^n)$. It is well known that the Hardy-Littlewood maximal function $Mf\notin L^1(\mathbb{R}^n)$ (if $f \ne 0$ a.e.), though there is a weak-type (1,1) bound for this maximal operator. More explicitly, we have a lower bound estimate $ Mf(x)\ge C|x|^{-n}$ (see e.g. <https://math.stackexchange.com/q... | https://mathoverflow.net/users/98145 | A simple question about the Hardy-Littlewood maximal function | No. Say we are in one dimension. Consider the function
$$ f(x) := \sum\_{m=2}^\infty 1\_{[m, m + \frac{1}{m \log^2 m}]}.$$
this function is (barely) in $L^1$ (because $\sum\_{m=2}^\infty \frac{1}{m \log^2 m}$ converges), but $M\_\phi f$ is (barely) outside of $L^1$ (basically because $\sum\_{m=2}^\infty \frac{1}{m ... | 7 | https://mathoverflow.net/users/766 | 251556 | 114,133 |
https://mathoverflow.net/questions/251562 | 1 | Let $(P,\leq)$ be a poset. The *interval topology* $\tau\_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
If $B$ is a complete Boolean algebra, is $(B,\tau\_i(B))$ Hausdorff?... | https://mathoverflow.net/users/8628 | Interval topology on complete Boolean algebras | No. You can't separate $0$ from $1$ in an atomless
Boolean algebra, such as the (complete)
Boolean algebra of regular open
subsets of $\mathbb R$.
The set of principal ideals and principal
filters form a subbasis of closed
sets for the topology on $B$, so a typical basic closed set has the form
$I\cup F$ where $I$ ... | 2 | https://mathoverflow.net/users/75735 | 251573 | 114,137 |
https://mathoverflow.net/questions/250950 | 1 | Let $V$ be a finite dimensional vector space over $ \mathbb{R}$. Let
\begin{equation} \left\langle\:,\:\right\rangle:\mbox{End}(V)\otimes\mbox{End}(V)\rightarrow \mathbb{R}\end{equation}
denote the pairing given by $\left\langle A,B\right\rangle = \mbox{Tr}(AB)$. We know that this pairing is $\mbox{GL}(V)$-invariant, s... | https://mathoverflow.net/users/nan | Conditions on $\beta$ under which the trace pairing restricted to $\mathfrak{so}(V,\beta)$ is positive (negative) definite | Yes to all three questions. By diagonalisation of forms, WLOG, $\beta$ is diagonal with $\pm 1$ on the main diagonal. Now let us just compute (and you can get an explicit formula for your transposition as well).
If $\beta$ is positive or negative definite, $so(V,\beta)$ is the set of skew-symmetric matrices. The trac... | 0 | https://mathoverflow.net/users/5301 | 251586 | 114,142 |
https://mathoverflow.net/questions/251555 | 5 | As the question title suggests, what is the role cohomology of coherent sheaves plays for SGA 4.5, étale cohomology? Why are they so important for the construction and establishing properties of étale cohomology? What are some of the main essences, intuitions, and theorems of the cohomology of coherent sheaves that are... | https://mathoverflow.net/users/nan | "Role" of cohomology of coherent sheaves in SGA 4.5, étale cohomology | You ask two different questions, I believe.
1) When Grothendieck invented étale cohomology, cohomology of coherent sheaves was already well known, thanks to Oka/Cartan's theorems A and B, Serre's FAC, Gaga, and the Serre-Grothendieck duality theorems). It was also well known (Weil, already) that the invention of some... | 9 | https://mathoverflow.net/users/10696 | 251587 | 114,143 |
https://mathoverflow.net/questions/251585 | 8 | Let $F$ be a number field such that $[F:\mathbb{Q}]=n$ and with ring of integers $O\_F$. Let's put $B=\operatorname{Spec } O\_F$, then an Arakelov divisor is an element of:
$$Div(X)\times \bigoplus\_\sigma \mathbb R[\sigma]$$
namely it can be written as
$$\bigg(\sum\_{\text{$\mathfrak p$ prime $\neq 0$}}n\_{\mathfrak ... | https://mathoverflow.net/users/47136 | Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings? | You would usually want the principal Arakelov divisors, i.e. those of the form $(\sum\_{\mathfrak{p}}{\rm ord}\_{\mathfrak{p}}(a), \sum\_\sigma -\log|\sigma(a)|)$ for $a\in F^\times$, to be cocompact in the group of degree $0$ divisors — the volume of the quotient should be the familiar product ${\rm Reg}(F)\cdot{\rm h... | 5 | https://mathoverflow.net/users/35416 | 251591 | 114,145 |
https://mathoverflow.net/questions/214767 | 13 | As we all know, the forgetful functor $\mathsf{Ab} \to \mathsf{CMon}$ from abelian groups to commutative monoids has a left adjoint, the **Grothendieck group**. I would like to **categorify** this construction.
For this, abelian groups should be replaced by symmetric monoidal categories in which every object is inver... | https://mathoverflow.net/users/2841 | Adding inverses to a symmetric monoidal category (Reference?) | Maybe you could be interested in the recent PhD thesis:
Une introduction élémentaire au 2-groupe de Grothendieck
by C. Drugmand, 2016, UCL, Louvain-la-Neuve.
<http://hdl.handle.net/2078.1/176774>
| 6 | https://mathoverflow.net/users/99044 | 251595 | 114,147 |
https://mathoverflow.net/questions/250865 | 3 | If $L$ is a Lie algebra over an algebraic closed field $K$ of characteristic zero, then all Cartan subalgebras are conjugated. Hence, they have all the same dimension. If $K$ is not algebraic closed but still of characteristic zero, then let $F$ the algebraic closure of $K$ and $H$ a Cartan subalgebra of $L$. Then $H\o... | https://mathoverflow.net/users/57804 | Unique dimension of Cartan subalgebras in modular Lie algebras | By a theorem of Premet ([MR](http://www.ams.org/mathscinet-getitem?mr=864177)) (which was later on again proven by Farnsteiner ([article](http://www.ams.org/journals/tran/2004-356-10/S0002-9947-04-03476-2) [MR](http://www.ams.org/mathscinet-getitem?mr=2058843))) the following theorem holds for a restricted Lie algebra ... | 1 | https://mathoverflow.net/users/57804 | 251615 | 114,152 |
https://mathoverflow.net/questions/251596 | 2 | If $h\_i:A\_i\to A\_{i+1}$ is a countable chain of morphisms in an abelian category $A$ that is AB3 then one can consider the (Bökstedt-Neeman) homotopy colimit of $A\_i$ in $D^b(A)$. This is a two-term complex $\coprod A\_i\stackrel{f}{\to} \coprod A\_i$ with the corresponding "components" of $f$ being $id\_{A\_i}$, $... | https://mathoverflow.net/users/2191 | On countable homotopy colimits in (the derived categories of) AB3 abelian categories | $\textrm{AB4}$ (and *a fortiori* $\textrm{AB3}$) is not enough, as it's not true for the opposite category of a module category, which is $\textrm{AB4}$.
Let $R$ be any ring, and consider the inverse system
$$\dots\to R[x]\stackrel{\theta}{\to}R[x]\stackrel{\theta}{\to}R[x]\stackrel{\theta}{\to}R[x],$$
of $R$-modules... | 1 | https://mathoverflow.net/users/22989 | 251622 | 114,155 |
https://mathoverflow.net/questions/251617 | 11 | Consider $g\_1$ and $g\_2$ two Riemannian metrics on a differentiable manifold $M$ of dimension **$n\ge 4$**. Suppose locally $g\_i=f\_i\sum\_{j=1}^ndx\_j^2$, where $f\_i:M\rightarrow \mathbb{R}$ are non negative functions. Suppose that both $g\_1$ and $g\_2$ have non positive sectional curvature.
Define the metric $\h... | https://mathoverflow.net/users/nan | Curvature of maximum of two riemannian metrics | Dimension 2 :
=============
The answer to your question is yes **in dimension 2**. I am not sure about higher dimension.
Set $f\_i=e^{2u\_i}$, so that $f=e^{2u}$ with $u=\max (u\_1,u\_2)$.
The Gauss curvature of a metric $e^{2v}(dx^2+dy^2)$ is given by $-e^{-2v}\Delta v$, so that each of the $u\_i$ is subharmonic... | 8 | https://mathoverflow.net/users/8887 | 251628 | 114,157 |
https://mathoverflow.net/questions/251636 | 9 | A well-known formula for the logarithm is given by
$$\log x = -\frac{\pi}{\operatorname{AGM}(a^2,b^2)}, \qquad x < 1$$
where AGM is the arithmetic-geometric mean, and $a$ and $b$ are given by
$$a = \sum\_{k\in\mathbb{Z}}x^{k^2}, \qquad
b = \sum\_{k\in\mathbb{Z}}x^{(k+1/2)^2};$$
Or, equivalently,
$$a = 1 + 2x ... | https://mathoverflow.net/users/99417 | Numerical coincidence? Why is $\sum(x^{k^2}) = \sum(x^{(k+1/2)^2})$ for $x = 0.8$? | The two values $a(0.8)$ and $b(0.8)$ appear "deceivingly" equal, but they actually are not!
Other "near-miss" values include
$$0<\theta\_3(0,0.9)-\theta\_2(0,0.9)<0.5\times 10^{-39}.$$
Let $f(x)=\theta\_3(0,x)-\theta\_2(0,x)$. The graph of $f(x)$, for values $0<x<1$ shows a global minimum at $x\_\*$ near $x=0.9$ (o... | 9 | https://mathoverflow.net/users/66131 | 251638 | 114,161 |
https://mathoverflow.net/questions/251624 | 5 | For a given set of numbers $A$, let $O^A$ be the hyperjump of $A$. It is possible to iterate inductively the hyperjump of a set, through the computable ordinals, in a way that the $\alpha$-th hyperjump is computably stronger than any $\beta$-th hyperjump for $\beta < \alpha$. It is even possible to keep iterating it th... | https://mathoverflow.net/users/14490 | On a characterization of the recursively inaccessible ordinals | Well assuming that $\lambda^A$ is always the first recursively admissible which is bigger than $\omega\_1^A$, which I think should be true, I think my question is after all not so interesting:
Either there is a largest recursively inaccessible smaller than $\omega\_1^A$, in which case $\lambda^A$ is a successor in th... | 2 | https://mathoverflow.net/users/14490 | 251645 | 114,165 |
https://mathoverflow.net/questions/251650 | 5 | The $n = 1$ case of Theorem 3.1 of Cohn and Elkies's paper *New upper bounds on sphere packings I* amounts to the inequality $f(0) \geq 1$ for all ('admissible') functions $f$ on $\mathbb{R}$ satisfying
* $\widehat{f}(0) = 1$;
* $f \leq 0$ outside of $[-1,1]$; and
* $\widehat{f} \geq 0$ everywhere.
I am interested ... | https://mathoverflow.net/users/26522 | Extremal functions for the 'packing density in dimension one' | You didn't make the precise assumptions on $f$ explicit, but even in a fairly general situation, your $\widehat{f}\_1$ is the only example that is supported by $[-1,1]$.
Given an $f$ with the stated properties, we can let $\widehat{g}=\widehat{f}^{1/2}$, so $f=g\*g$, and if $\textrm{supp}\, f\subseteq [-1,1]$, then [... | 3 | https://mathoverflow.net/users/48839 | 251662 | 114,169 |
https://mathoverflow.net/questions/251651 | 10 | Let $X=\{x\_1,...,x\_k\}\subset E^n$ be a finite subset in the Euclidean $n$-space, $r>0$ and $B(x\_i,r)$ are open balls of radius $r$ centered at the points $x\_i\in X, i=1,...,k$. Suppose that
$$
\bigcap\_{i=1}^k B(x\_i,r)\ne \emptyset.
$$
Is it true that the convex hull of $X$ is contained in
$$
\bigcup\_{i=1}^k B(... | https://mathoverflow.net/users/21684 | covering convex sets by round balls | Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m\_i$ in $x\_i$, $\sum m\_i=1$, $y=\sum m\_i x\_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m\_i |p-x\_i|^2$, $p\in E^n$, just because $I(p)=I(y)+|p-y|^2$. In particular, it can not happen that all distances $|y-x\_... | 14 | https://mathoverflow.net/users/4312 | 251666 | 114,172 |
https://mathoverflow.net/questions/251627 | 14 | I am currently teaching a topics course where I talk about some discrete groups acting properly. A student asked a very basic question that stumped me: what is the precise relationship between proper discontinuity and existence of a fundamental domain? Namely, let $G$ be a discrete group acting on a topological space $... | https://mathoverflow.net/users/39348 | Proper discontinuity and existence of a fundamental domain | I will assume that you are interested in group actions on connected manifolds: In the case of more general spaces it is not even completely clear what a fundamental domain means since an element of finite order can fix a nonempty open subset.
**Definition.** Let $M$ be a manifold (in any category you like, DIFF, PL o... | 14 | https://mathoverflow.net/users/21684 | 251680 | 114,178 |
https://mathoverflow.net/questions/251646 | 10 | For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \det(C) \geq \det(A+C) + \det(B+C).$$
The first is, of course, Minkowski's determinant inequality.
I'm not sure whethe... | https://mathoverflow.net/users/99418 | Reverse Minkowski (and related) Determinant Inequalities | Inequality ($\star\star$) essentially follows from the original Minkowski plus an implication of Lidkskii's inequality (Fiedler's inequality, noted below).
$\newcommand{\da}{\downarrow} \newcommand{\ua}{\uparrow}$
Assume $C$ is strictly positive definite (otherwise, $\det C=0$ rendering ($\star\star$) trivial), and l... | 7 | https://mathoverflow.net/users/8430 | 251684 | 114,180 |
https://mathoverflow.net/questions/251685 | 6 | In Hardy-Littlewood's 1923 paper *"Some problems of 'Partitio Numerorum' III"* it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\chi)$ have $\Re(s)<3/4-\varepsilon$), that for all $k\geqslant 3$, when $n\to\infty$ through the integers with same parity than $k$:
... | https://mathoverflow.net/users/74026 | Vinogradov's method for sums of more than three primes | Your question is concerned with the so-called *Waring-Goldbach problem*. A classic in this topic is Hua Lo Keng's book, Additive theory of prime numbers (Translations of Mathematical Monographs, 13, American Mathematical Society, Providence, R.I. 1965), which focuses on how large the number of summands $s$ should be in... | 3 | https://mathoverflow.net/users/11919 | 251687 | 114,181 |
https://mathoverflow.net/questions/251688 | 29 | How much should an average mathematician not working in an area like logic, set theory, or foundations know about the foundations of mathematics?
The thread [Why should we believe in the axiom of regularity?](https://mathoverflow.net/questions/219590/why-should-we-believe-in-the-axiom-of-regularity) suggests that man... | https://mathoverflow.net/users/99445 | How much should the average mathematician know about foundations? | The answer is essentially the same as *how much should the average mathematician know about combinatorics*? Or *group theory*? Or *algebraic topology*? Or any broad area of mathematics... It's good to know some, it's always helpful to know more, but only really need the amount that is relevant to your work. Perhaps a s... | 26 | https://mathoverflow.net/users/2000 | 251693 | 114,183 |
https://mathoverflow.net/questions/251695 | 3 | It's well known that the Ornstein-Uhlenbeck semigroup defined by
$$
P\_tf(x)=\int\_{\mathbb{R}}f\left(xe^{-t}+\sqrt{1-e^{-2t}}z\right)\frac{e^{-z^2/2}}{\sqrt{2\pi}}\,dz
$$
is **not** strongly continuous on the space $C(\mathbb{R})$ of continuous functions on $\mathbb{R}$ with the supremum norm. I was wondering if the O... | https://mathoverflow.net/users/4047 | Strong continuity of the Ornstein-Uhlenbeck operator | Nope. To see this, let $X\_t(x)$ denote the OU process at time $t>0$ with initial condition $x$; set $f(x) = g(x) (1+|x|^k)$ where $g \in C\_b(\mathbb{R})$; and consider:
\begin{align\*}
\frac{P\_t f(x) - f(x)}{1+|x|^k} &= \frac{E\{ g(X\_t(x)) (1+|X(t)|^k) - g(x) (1+|x|^k) \}}{1+|x|^k} \\
&= \frac{E\{ g(X\_t(x)) (1+|X\... | 2 | https://mathoverflow.net/users/64449 | 251703 | 114,186 |
https://mathoverflow.net/questions/251697 | 3 | I'm playing with the proof assistant Coq (which assumes ex falso quodlibet, but neither double negative or tertium non datur) and can easily prove (A∨¬A) ⇒ (¬¬A→A). I can not prove (¬¬A→A) ⇒ (A∨¬A), and neither can "tauto" of Coq (although the statement is true). tauto can prove (¬¬A→A) ⇔ ((¬A→A)→A) (although I don't h... | https://mathoverflow.net/users/11504 | Glivenko's theorem implying all "weakly true" statements created equal? | No.
For question 1, let
$$P\_1 = A \vee \neg A,$$
$$P\_2 = \neg\neg A \vee \neg A.$$
Their double negations are both tautologies, but $P\_1$ and $P\_2$ are not equivalent.
For question 2, let
$$P\_1 = (A \wedge \neg\neg B) \vee (A \wedge \neg B) \vee \neg A,$$
$$P\_2 = (\neg\neg A \wedge B) \vee (\neg\neg A \wedge... | 3 | https://mathoverflow.net/users/nan | 251711 | 114,189 |
https://mathoverflow.net/questions/251712 | 1 | What is the general class of manifolds isometrically embedded in $R^3$ for which there exists some point in $R^3$ such that any plane through the point yields a geodesic where it intersects the manifold? Two immediate examples are: a plane, with respect to any point; a sphere, with respect to its center point.
| https://mathoverflow.net/users/99458 | Class of embedded 2D manifolds whose geodesics are coplanar with a point | Only spheres and planes have this property.
I assume that the surface is sufficiently smooth ($C^2$).
Every spatial curve has the so-called principal normal at every point of non-zero curvature, this is the acceleration direction for the unit speed motion along the curve.
The principal normal to a geodesic coincides ... | 4 | https://mathoverflow.net/users/98590 | 251722 | 114,194 |
https://mathoverflow.net/questions/251713 | 8 | Are there analogues of the $j$-invariant for higher genus curves?
Of course for genus $g \ge 2$ there will have to be at least $3g-3$ invariants. Are there "canonical choices" for these invariants, viewed as functions on the coarse moduli schemes? (ideally with relatively nice arithmetic properties)
I would appreci... | https://mathoverflow.net/users/88840 | Invariants of higher genus curves | For $g > 2$ you're not going to be able to write such invariants as *global* functions on the moduli space.
In general for an algebraic variety $X$, there is a canonical morphism $a \colon X \to \mathrm{Spec}(H^0(X,\mathcal O\_X))$, the *affinization* of $X$. Points $x$ and $y$ of $X$ are separated by some global re... | 14 | https://mathoverflow.net/users/1310 | 251730 | 114,197 |
https://mathoverflow.net/questions/251728 | 5 | We have the following theorem of Baker:
**Theorem 1.** Let $\alpha\_1, \ldots, \alpha\_m \in \mathbb{C}$ be algebraic numbers $\neq 0, 1$ such that $\log \alpha\_1, \ldots, \log \alpha\_m$ are linearly independent over $\mathbb{Q}$. Then $\log \alpha\_1, \ldots, \log \alpha\_m$ are linearly independent over the algeb... | https://mathoverflow.net/users/nan | Linear independence of p-adic logarithms (analog of Baker's theorem) | Yes. For linear independence, the result goes back to A. Brumer in connection with the Leopoldt conjecture. Brumer, A. "On the units of algebraic number fields", Mathematika 14 (1967) 121–124
For lower bounds on linear forms, there was an early version due to van der Poorten that I think had some problems and then a ... | 7 | https://mathoverflow.net/users/2290 | 251732 | 114,198 |
https://mathoverflow.net/questions/251704 | 2 | Let $G$ be a finite group and $M$ be a nontrivial proper subgroup of $G$ with the following conditions:
a) If $H$ is a subgroup of $G$ such that $M\lneqq H\lneqq G$, then $H$ contains at least one minimal subgroup of $G$ say $L$, such that $M\cap L=1$.
b) If $K$ is a subgroup of $G$ such that $M\cap K=1$ then $K\co... | https://mathoverflow.net/users/97247 | Maximal Subgroups | I remember answering a very similar (possibly the same) question recently but I cannot find it.
Let $G$ be a central product of a cyclic group $M$ of order $4$ and the dihedral group $D\_8$ of order $8$. So $|G|=16$. (In fact you get an isomorphic group if you replace $D\_8$ by $Q\_8$. A central product of $C\_4$ wi... | 5 | https://mathoverflow.net/users/35840 | 251736 | 114,200 |
https://mathoverflow.net/questions/251745 | 0 | let $X$ be a (complex) Banach space, and $\{x\_n\}$ is a sequence in $X$. Suppose that for any $f\in X'$, $$\sum\_{n=1}^\infty |f(x\_n)|<\infty.$$ Show that there exists a constant $\mu>0$ such that $$\sum\_{n=1}^\infty |f(x\_n)|\leq \mu ||f||.$$
Here, $X'$ is the Banach space consisting of all linear bounded functio... | https://mathoverflow.net/users/99478 | On the dual of Banach space | You can prove the desired as follows. Consider the map from $X'$ to $\ell\_1$ defined by $f\mapsto \{f(x\_n)\}\_{n=1}^\infty$. Show that it satisfies the assumptions of the Closed Graph Theorem, conclude that it is a continuous operator. The desired inequality follows.
| 1 | https://mathoverflow.net/users/85406 | 251750 | 114,205 |
https://mathoverflow.net/questions/251740 | 6 | What is an example of a lattice $(L,\leq)$ that is uniquely complemented, but not Boolean?
| https://mathoverflow.net/users/8628 | Uniquely complemented but not Boolean | As mentioned in the comments, there are and this follows from Dilworth's construction of free uniquely complemented lattices. A simpler construction was later given by Chen and Grätzer [[On the construction of complemented lattices](http://www.ams.org/mathscinet-getitem?mr=232715), J. Algebra 11 (1969), 56–63]. There i... | 9 | https://mathoverflow.net/users/2000 | 251760 | 114,208 |
https://mathoverflow.net/questions/251749 | 1 | In complex analysis, almost all of the book on Nevanlinna theory will mention that Ostrowski first constructed an example of mermorphic function without Julia direction.
Furthermore, recently, from Segal's book "Nine introduction in complex analysis", I know that not only did Ostrowski construct the example, but als... | https://mathoverflow.net/users/11966 | English reference for Ostrowski's theorem for Julia exceptional function | Ostrowski's knew form the theory of normal families that if $f(z)$ is a Julia exceptional function, it has to be a meromorphic function of order zero. Thus it can be expressed as
$$f(z)=z^m \frac{\prod (1-\frac{z}{a\_\alpha})}{\prod (1-\frac{z}{b\_\beta})}$$
His result was that such a $f(z)$ is Julia exceptional if... | 1 | https://mathoverflow.net/users/43108 | 251766 | 114,210 |
https://mathoverflow.net/questions/251779 | 6 | It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this is true ?
For example, is it true that for all reflexive complete locally convex topological vector spaces, closed co... | https://mathoverflow.net/users/99246 | Closed convex bounded sets are weakly compact for which spaces? | It is well-known that a Hausdorff locally convex space is semi-reflexive (i.e., the canonical map into its bidual is surjective) if and only if every weakly closed bounded set is weakly compact. This is proposition 23.18 in *Introduction to Functional Analysis* of Meise and Vogt.
(It is mainly a consequence of Alaogl... | 6 | https://mathoverflow.net/users/21051 | 251789 | 114,215 |
https://mathoverflow.net/questions/251515 | 1 | Does the following statement hold?
Let $p$ be a rational prime. For each $x\in\prod\_{\ell}{\mathbb Z}\_{\ell}$, there exist a natural number $M\_x$ such that $x$ lies in the closure of $\{e\_1p^{n\_1}+e\_2p^{n\_2}+\cdots+e\_{M\_x}p^{n\_{M\_x}}: \text{$n\_i\ge 0$ and $e\_i\in\{-1,0,1\}$ for each $i$}\}$ in $\prod\_{... | https://mathoverflow.net/users/4948 | Convergence in a product of $p$-adic groups | The answer is negative and we can tell this just by looking at $\mathbb Z\_p$.
For each $M$, the set of sums of $\leq M$ terms of the form $\pm p^n$ for arbitrary nonnegative $n$ is closed in $\mathbb Z\_p$. Because convolutions of closed sets are closed, it is sufficient to prove this for $\{\pm p^n\} \cup \{0\}$, w... | 2 | https://mathoverflow.net/users/18060 | 251791 | 114,217 |
https://mathoverflow.net/questions/251753 | 0 | Question with three parts:
a. Is there a theorem which states an upper limit function for rate of the decay of $\int\_x^\infty f(y)dy$, as $x\rightarrow\infty$?
b. Assume $f(y)$ is positive, real, and restriceted to the positive-real y's, bounded from above by some constant 'a' and $\lim\_{y\rightarrow\infty}[f(y... | https://mathoverflow.net/users/37545 | Upper limit function for the decay of $\int_x^\infty f(y)dy$, provided $f(y)$ is analytic, when $x\rightarrow\infty$? | $f(x):=\frac{\sin^2 x}{x^2}$ is entire, bounded on $\mathbb R$, and $\int\_x^\infty f(y)\ dy$ decreases as $cx^{-1}$.
| 4 | https://mathoverflow.net/users/75422 | 251792 | 114,218 |
https://mathoverflow.net/questions/251744 | 1 | Let $X$ be a smooth, projective variety over an algebraically closed field $k$ (of characteristic zero), $B$ a connected, noetherian scheme (possibly non-reduced) and $U$ an open subscheme of $X \times\_k B$ such that for every closed point $b \in B$, the complement of $U\_b:=U \cap (X \times \{b\})$ is of codimension ... | https://mathoverflow.net/users/45397 | Extending locally free sheaves and compatibility with fibers | I am just posting my comment as an answer.
In fact, the natural homomorphisms $\mathcal{O}\_{X\_o}\to (j')\_\*\mathcal{O}\_{U\_o}$ and $\mathcal{O}\_{X\times B} \to j\_\*\mathcal{O}\_U$ are both isomorphisms. This is the type of result discussed in EGA IV, Sections 5.9 and 5.10. If Grothendieck does not convince you... | 1 | https://mathoverflow.net/users/13265 | 251795 | 114,219 |
https://mathoverflow.net/questions/251802 | 6 | Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. For a good prime $p$, define $\theta\_{E}(p)$ by
$$\cos\theta\_{E}(p)=\frac{p+1-N\_{p}(E)}{2\sqrt{p}}\quad (0\leq \theta\_{E}(p)\leq \pi).$$
I wonder if the function $\theta\_{E}$ is injective?
| https://mathoverflow.net/users/99510 | Question on the Sato-Tate conjecture | No. If $E\_p$ is a supersingular elliptic curve and $p>3$ then trace of Frobenius on $E\_p$ is zero, so $\theta\_E(p)=\pi/2$.
By a result of Elkies any elliptic curve over $\mathbb{Q}$ has supersingular reduction in infinitely many primes, so for infinitely many $p$ this function takes the same value.
| 12 | https://mathoverflow.net/users/39304 | 251803 | 114,220 |
https://mathoverflow.net/questions/251762 | 2 | Let $X$ be a finite subset of real numbers. Let $G$ be the collection of all non empty subsets of $X$ and $G\_{0}$ be the collection of all singleton subsets of $X$.
We define two maps $r,s:G \to G\_{0}$ with $$r(A)=\text{The singletone consisting of Maximum} \;\; of\;\; A$$ and $$s(A)=\text{The singleton consisting... | https://mathoverflow.net/users/36688 | Certain groupoid and its $C^{*}$ algebra | I will explain how I ended up with the description given in my comment above.
Let $\{1, \dots ,n \}$ be your finite subset of $\mathbb{R}$ (only their order is important, we do not really care about the precise values).
A first remark is that your semi-groupoid is in fact a category: $\{i\}$ is an identity from $i$... | 1 | https://mathoverflow.net/users/22131 | 251805 | 114,221 |
https://mathoverflow.net/questions/251820 | 6 | There's and old and extensively studied question about characterisation of fundamental groups of smooth compact Kähler manifolds. Restrictions imposed by Kählerness are somewhat fragile, and if we drop some properties and go aside a bit, every finitely presented group can be realized as a $\pi\_1$ of 1) symplectic comp... | https://mathoverflow.net/users/81055 | Restrictions on $\pi_1(X)$ of geometric origin (Kähler groups as example) | Some properties of Kähler groups are retained by other classes of closely related groups, such as fundamental groups of smooth, quasi-projective varieties, or fundamental groups of Sasakian manifolds.
For instance, if $M$ is a smooth, quasi-projective variety, then the first characteristic variety $V^1\_1(M)$, i.e.,... | 13 | https://mathoverflow.net/users/17846 | 251822 | 114,229 |
https://mathoverflow.net/questions/251773 | 7 | Let $P$ be a finite poset and let $\,\mathcal{C}=(C\_1,\ldots,C\_\ell)\,$ be its decomposition into chains. We can define
$$
f(\mathcal{C}) = |C\_1|! \, \cdots \,|C\_\ell|!
$$
and ask for what $\mathcal{C}$ we have $f(\mathcal{C})$ maximal. Roughly speaking, the smaller is $\ell$ and the less evenly distributed are $|C... | https://mathoverflow.net/users/4040 | Entropy of chain decompositions of posets | The answer is **no**. Take a chain $a\_1>\dots>a\_{100}$ and add two elements $b>a\_{51}$ and $c<a\_{50}$ (with no relations not implied by these ones). The maximum of $f(\mathcal C)$ is achieved on the decomposition $\{a\_1,\dots,a\_{100}\}\sqcup\{b\}\sqcup\{c\}$, while the maximal antichain has cardinality 2 (due to ... | 3 | https://mathoverflow.net/users/17581 | 251824 | 114,231 |
https://mathoverflow.net/questions/251751 | 6 | Let $W(B\_n)$ be a Weyl group of type $B\_n$ and $SBT (n)$ the set of standard bitableaux of size $n$. Similar to Robinson-Schensted correspondences, I know that there exists a map $W(B\_n) \to SBT (n) \times SBT (n)$ such that the image of $W(B\_n)$ is the set of the same shape pairs of standard bitableaux.
But, I ... | https://mathoverflow.net/users/89288 | Correspondence between $SBT (n)$ and $W(B_n)$ | Before getting into my answer, let me explain what I understand by a bitableau of size $n$ (I hope it is the same as what you mean). This is a pair of tableaux $(P, P')$, where each of the integers $1,\dotsc,n$ occurs exactly once, and moreover, the numbers increase along rows and columns in both $P$ and $P'$. The shap... | 3 | https://mathoverflow.net/users/9672 | 251826 | 114,232 |
https://mathoverflow.net/questions/251821 | 2 | In the studies of the calculus of variation, a map $f:M\to N$ said to be harmonic if it is a critical point of the Dirichlet energy function. i. e.
\begin{align}
E:C^\infty(M,N)&\longrightarrow \Bbb R\\
f&\mapsto E(f)
\end{align}
In general, a point $p$ in $M\_1$ is a critical point of $f:M\_1\to M\_2$ if the diffe... | https://mathoverflow.net/users/90655 | Manifold structure of the set of all smooth functions between two smooth manifolds! | You can find the smooth case in *The Convenient Setting of Global Analysis* (by Andreas Kriegel & Peter Michor), Chapter IX, Manifolds of Mappings.
If you are also interested in the case $k<\infty$: $C^k(M,N)$ (where $M$ is compact and $N$ Riemannian) with the $C^k$-compact open topology is a $C^\infty$-Banach manifo... | 4 | https://mathoverflow.net/users/97669 | 251828 | 114,233 |
https://mathoverflow.net/questions/251718 | 4 | Let $R$ be a finite ring and $F$ be an algebraically closed field in which $|R|$ is invertible. Does there exists an $F$-valued character $\chi$ of $(R, +)$ such that every character $\psi$ is of the form $\psi(a) = \chi(ab)$ for some $b \in R$? If not, does the statement hold when $R$ is commutative (or under any othe... | https://mathoverflow.net/users/70959 | Action of certain endomorphisms on Pontriyagin dual | Call a character $\chi$ *(left) generating* if every character is of the form $\psi(a)=\chi(ab)$ for some $b\in R$. It turns out that, when $R$ is finite, a character is left generating if and only if it is right generating (every character has the form $\psi(a) = \chi(ba) $ for some $b\in R$).
The answer to your qu... | 4 | https://mathoverflow.net/users/10266 | 251834 | 114,236 |
https://mathoverflow.net/questions/251735 | 5 | In [[CEL84, Theorem 1.1, p.489]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf), Crandall, Evans, and Lions give three equivalent definitions of *viscosity solution*.
As the authors note, the first two are "more appealing in some respects and more convenient for... | https://mathoverflow.net/users/nan | On the 'usefulness' of the 'original' definition of viscosity solution | **(i).** The semijet definition [[CEL Theorem 1.1 (i)]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf) is the natural way to derive comparison principles for PDEs using, e.g., the Crandall-Ishii lemma [[CIL, Theorem 3.2]](http://www.ams.org/journals/bull/1992-27-01... | 1 | https://mathoverflow.net/users/35874 | 251839 | 114,237 |
https://mathoverflow.net/questions/251851 | 3 | Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local).
If $p$ is a height one prime, then the localization $R\_p$ is a dvr, hence the maximal ideal $pR\_p$ is principal, generated by a uniformizer $\pi$.
It's easy to see we can pick $\pi \in p$. Similarly, if $\pi' \i... | https://mathoverflow.net/users/16857 | uniqueness of uniformizers | This is the same as asking that $p$ is principal. In one direction, if $p = (\pi)$, then take $\pi$ to be the uniformizer.
In the reverse direction, suppose $\pi$ has the stated property. I claim that $p = (\pi)$. Let $f$ be a nonzero element of $p$; we must show that $f$ is a multiple of $\pi$. Since $R$ is a noeth... | 4 | https://mathoverflow.net/users/297 | 251854 | 114,241 |
https://mathoverflow.net/questions/251833 | 6 | I'm interested in the following symmetric functions $s\_k: \mathbb{R}\_+^{k}\mapsto\mathbb{R\_+}$:
\begin{align\*}
s\_k(x\_1,x\_2,\dots, x\_k)&=
\int\limits\_{0=t\_0<t\_1<\dots<t\_{k-1}<t\_k=1} e^{-(t\_1-t\_0) x\_1}
e^{-(t\_2-t\_1)x\_2}\dots e^{-(t\_k-t\_{k-1})x\_k}dt\_1 \dots dt\_{k-1} \\
&=\sum\_{i=1}^k \frac{e^{-x\... | https://mathoverflow.net/users/5784 | Symmetric functions arising from continuous-time Markov chains | Let the random variable $t\_{k-1}$ be the sum of the holding times in states $1$ up to state $k-1$. Note that $t\_{k-1}$ is a [hypoexponential random variable](https://en.wikipedia.org/wiki/Hypoexponential_distribution). Let $f\_{k-1}$ denote the PDF of this random variable. From the probabilistic interpretation given ... | 2 | https://mathoverflow.net/users/64449 | 251861 | 114,243 |
https://mathoverflow.net/questions/251856 | 0 | For real numbers, we know that any monotonic bounded sequence converges to a finite limit. Does this generalize to sequences of operators?
More formally, I have a sequence of operators $\{A\_n\}\_{n=1}^{\infty}$ where each $A\_n: \ell\_1 \to \ell\_1$ and $\|A\_n\|\_1 \leq 1$. I know from the Banach-Alaoglu theorem th... | https://mathoverflow.net/users/51134 | Monotone convergence theorem for operators in the weak operator topology | You've clarified in comments that $\le$ means that the operators $A\_{n+1} - A\_n$ are positivity preserving. So for any nonnegative $x \in \ell^1$, we have $A\_1 x \le A\_2 x \le \dots$ pointwise. Since $A\_1 x \in \ell^1$ and the sequence $\{A\_n x\}$ is $\ell^1$-bounded, the monotone convergence theorem implies that... | 5 | https://mathoverflow.net/users/4832 | 251863 | 114,244 |
https://mathoverflow.net/questions/251847 | 1 | Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X\_R:=X \times\_k \mathrm{Spec}(R)$ such that its restriction to the generic fiber has finite homological dimension but the restriction to the ... | https://mathoverflow.net/users/45397 | Homological dimension of pure coherent sheaves and specialization | Take for $X$ the plane curve $X^3=Y^2T$, and for $F$ the ideal sheaf $(X-\pi ^2T,Y-\pi ^3T)$. Its restriction to the generic fiber is the ideal of a smooth point, hence is an invertible sheaf, while its restriction to the special fiber is the ideal of the singular point $(0,0,1)$, which has infinite homological dimensi... | 2 | https://mathoverflow.net/users/40297 | 251866 | 114,245 |
https://mathoverflow.net/questions/251850 | 3 | Let $n=\prod\_{p|n}p^{\alpha\_p}$. While considering arithmetic functions of the projection $$\textrm{rad}\_k(n)=\prod\_{p|n}p^{\textrm{min}(\alpha\_p,k)},$$ a basic question has arisen that is more difficult to answer than I had supposed.
Let $a,f:\mathbb{N}\to\mathbb{C}$ and define $$Af(n)=\sum\_{d|n}a(d)f(d)$$
... | https://mathoverflow.net/users/10980 | Projective summation over divisors: must these functions be multiplicative? | It's not the case when $k=2$ even: other solutions are, for any integer $j\ge2$,
$$
a(n) = \begin{cases}
1, &\text{if } n=1, \\
-1, &\text{if } n=j, \\
0, &\text{otherwise.}
\end{cases}
$$
In particular, this function is not multiplicative when $j=6$. Indeed, there seem to be lots of solutions that involve setting $a(p... | 2 | https://mathoverflow.net/users/5091 | 251868 | 114,246 |
https://mathoverflow.net/questions/251877 | 4 | Is there any combinatorial interpretation or bijective proof for this identity
$$2C\_n=4{2n \choose n}-{2n+2 \choose n+1}$$
where $C\_n$ is the sequence of Catalan numbers?
| https://mathoverflow.net/users/83921 | Is there a combinatorial interpretation or bijective proof for this Catalan number identity? | There is an obvious bijective proof of the identity
$$ 2 \binom{2n}{n} + 2 \binom{2n}{n + 1} = \binom{2n+2}{n+1}$$
and also a bijective proof of
$$ 2\binom{2n}{n} - 2 \binom{2n}{n+1} = 2C\_n,$$
see the paragraph "second proof" [in the wiki page](https://en.wikipedia.org/wiki/Catalan_number#Second_proof). By combining t... | 6 | https://mathoverflow.net/users/21724 | 251880 | 114,249 |
https://mathoverflow.net/questions/251885 | 4 | In the book of Iskovskikh and Prokhorov it seems not known wether the $V\_1$, an hypersurface of degree $6$ in the weighted projective space $\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any progress since then ?
| https://mathoverflow.net/users/76193 | Rationality of $V_1$ fano threefold | You mean degree 6 in $\mathbb{P}(3,2,1,1,1)$. It is not rational, and its birational automorphisms are biregular. This has been proved by M. Grinenko, *Mori structures on a Fano threefold of index 2 and degree 1*, Proc. Steklov Inst. Math. 246 (2004), 103-128. The russian version is available [here](http://www.mathnet.... | 8 | https://mathoverflow.net/users/40297 | 251886 | 114,252 |
https://mathoverflow.net/questions/251835 | 13 | Let $p$ be a prime and let $M$ be an $n \times m$ matrix with integer entries such that $M\vec{v} \not\equiv \vec{0} \text{ (mod }p\text{)}$ for any column vector $\vec{v} \neq \vec{0}$ whose entries are $0$ or $1$.
Is there a row vector $\vec{x}$ with integer entries such that no entry of $\vec{x}M$ is $0 \text{ (mo... | https://mathoverflow.net/users/99127 | Is there a row vector $x$ with integer entries such that no entry of $xM$ is $0 \text{ (mod }p\text{)}$? | It seems that my previous answer was completely wrong, and that the fact is **true** for all nonzero integer (not necessarily prime) $p$.
Let $M=[m\_{ij}]\_{i\leq n,\;j\leq m}$. If there is no $x$ satisfying the requirements (we regard all entries as integers, not residues!), then one of the sums of the form $\sum\_{... | 10 | https://mathoverflow.net/users/17581 | 251898 | 114,257 |
https://mathoverflow.net/questions/239103 | 5 | Terminology and context
-----------------------
(This should all be standard, but is recalled because terminology sometimes varies, and also to put the question into perspective.)
A partially ordered set is called **well-founded** iff it has no infinite decreasing sequence. It is called **well-partially-ordered** (... | https://mathoverflow.net/users/17064 | Does the rank (=height) of a well partial order bound its type (=length, =stature)? | $\DeclareMathOperator{\rk}{rk}$If I’m not mistaken, there’s a rather easy crude bound: $o(P) < (\omega \cdot (\rk P))^+$, where $(-)^+$ is cardinal successor.
Since $o(P)$ is the order-type of some linearisation, it’s clear that $|o(P)| = |P|$, and so $o(P) < |P|^+$. So if we can bound the *cardinality* of $P$ based ... | 4 | https://mathoverflow.net/users/2273 | 251903 | 114,260 |
https://mathoverflow.net/questions/251931 | 10 | If k is a non-discrete topological field, we can define an analytic space over k just like complex analytic spaces over $\mathbb{C}$. If you replace "complex analytic space" and "complex algebraic variety" with "analytic space over $k$" and "algebraic variety over $k$", respectively, under what conditions on $k$ does G... | https://mathoverflow.net/users/83073 | Does GAGA hold over other topological fields? | If $k$ is a field that is complete with respect to some ultrametric valuation, then there is the "GAGR" (i.e. *géométrie algébrique et géométrie rigide*) theorem. A succinct explanation (in French, without proof) is:
>
> Jarraud, Pierre. [*À propos de G.A.G.R..*](http://eudml.org/doc/91922) Groupe de travail d'anal... | 10 | https://mathoverflow.net/users/17907 | 251933 | 114,267 |
https://mathoverflow.net/questions/251916 | 8 | For a given positive integer $n$, I need to learn the number of $n\times n$ matrices of nonnegative integers with the following restrictions:
1. The sum of each row and column is equal to $3$.
2. Two matrices are considered equal if one can be obtained by permuting rows and/or columns.
For example, for $n=2$ there ... | https://mathoverflow.net/users/97013 | Number of all different $n\times n$ matrices where sum of rows and columns is $3$ | <http://oeis.org/A001501> has an enumeration of binary matrices with row and column sums equal to 3. If n is the order of the matrix, the count grows like n^{3n}. You want classes up to row and column permutations, so your number will grow like n^n.
There are some details to handle, but most of your cases reduce to t... | 3 | https://mathoverflow.net/users/3402 | 251934 | 114,268 |
https://mathoverflow.net/questions/251950 | 6 | Let $X$ be a smooth projective surface over $\mathbb{C}$. Then there is the exponential sheaf sequence:
$$
0 \rightarrow \mathbb{Z} \rightarrow \mathscr{O}\_X \rightarrow \mathscr{O}\_X^\times \rightarrow 0
$$
with the map from $\mathscr{O}\_X$ to $\mathscr{O}\_X^\times$ given by $f \mapsto \exp(f)$. Since the sequence... | https://mathoverflow.net/users/56878 | Algebraic vs. homological equivalence for curves on a smooth complex projective surface | Super vast generalisation: for divisors on a smooth projective variety over an algebraically closed field of *any* characteristic, the notions of algebraic, homological (for any Weil cohomology theory), and numerical equivalence agree (up to torsion).
**Remark.** Recall: the group of cycles $\alpha \sim\_{\text{alg}}... | 11 | https://mathoverflow.net/users/82179 | 251958 | 114,273 |
https://mathoverflow.net/questions/251959 | 2 | Let $ex(n,H)$ denote the maximum number of edges of a graph on $n$ vertices not containing a copy of $H$. Let $ex(n,m,H)$ denote the maximum number of edges of a bipartite graph with parts' sizes $m$ and $n$ not containing a copy of $H$.
I'm interested in upper bound on $ex(n, n, C\_4)$. It is easy to show with probab... | https://mathoverflow.net/users/97131 | Maximum number of edges in bipartite graph without cycles of length 4 | I find out that this problem is a special case of Zarankiewicz problem and it was solved by István Reiman in 1958 (thanks to Oliver Krüger for correction). The answer is $ex(n,n,C\_4)\sim n^{3/2}$.
| 2 | https://mathoverflow.net/users/97131 | 251962 | 114,274 |
https://mathoverflow.net/questions/251800 | 5 | It is known that if a Banach space is reflexive and separable, its unit ball is weakly metrizable.
My question is about the generalization of this property :
1) Is it true that for all reflexive separable locally convex space, bounded sets are weakly metrizable?
2) If that's true, is there a way to explicitly ... | https://mathoverflow.net/users/99246 | Are bounded sets always weakly metrizable in reflexive separable spaces? | No. Let $I$ be an index set with the cardinality of the continuum. Endow $X=\mathbb R^I$ with the product topology. According to (a particular case of) the Hewitt-Marczewski-Pondiczery theorem (which is 2.3.15 in Engelking's *General Topology*) $X$ is separable. Moreover, it is semi-reflexive (by Tychonov) and barrelle... | 6 | https://mathoverflow.net/users/21051 | 251966 | 114,276 |
https://mathoverflow.net/questions/251976 | 1 | Let's say we have a sequence $a\_n$ that is defined for all $n\in\mathbb{Z}$
and i want to work with its GF $$A(z)=\sum\_{n\in\mathbb{Z}}a\_nz^n$$
But there are some problems with convergence. For example for $a\_n=1$:
$$A(z)=\cdots\frac{1}{z^3}+\frac{1}{z^2}+\frac{1}{z}+1+z+z^2+\cdots=\frac{1}{1-\frac{1}{z}}+\frac{z... | https://mathoverflow.net/users/83921 | How to work with this power series? | Yes, you may consider them as a module over the ring of polynomials. In other words, you may sum up such series and multiply them by polynomials. Also, you may differentiate such series, and usual rules work. For example, your series $\sum\_{n\in \mathbb{Z}} z^n=A(z)$ satisfies $(1-z)A(z)=0$. An application of this mod... | 7 | https://mathoverflow.net/users/4312 | 251977 | 114,277 |
https://mathoverflow.net/questions/245493 | 0 | Let $x\_{k+1} = \frac{x\_k^{d\_k}+1}{x\_{k-1}}$, $k \in \mathbb{Z}$, where $d\_{k+2} = d\_k \in \mathbb{Z\_{>0}}$. Let $b=d\_1$ and $c=d\_2$. Define the cluster algebra $A = A(\left( \begin{matrix} 0 & b \\ -c & 0 \end{matrix} \right))$ to be the algebra generated by all $x\_k$. Is the following set $B$ a totally posit... | https://mathoverflow.net/users/11877 | Canonical basis of cluster algebras | You should look at [Positivity and canonical bases in rank 2 cluster algebras of finite and affine types](https://arxiv.org/abs/math/0307082) by Sherman and Zelevinsky where they are able to construct a canonical basis explicitly for *finte type* and *affine type* (i.e. $bc < 4$ and $bc = 4$ respectively). Your $B$ is ... | 1 | https://mathoverflow.net/users/51668 | 251984 | 114,282 |
https://mathoverflow.net/questions/251990 | 1 | I have the functional equation
$$ f(x+g(x)y)=f(x)+f(g(0)y)-f(0) $$
where
* $f$ is positive, monotone increasing and continuous
* $g$ is continuous and positive
* The domain of both functions is a closed interval that includes 0.
The obvious solutions are:
* $g$ constant and $f$ linear
* $f$ constant and $g$ arb... | https://mathoverflow.net/users/74274 | Cauchy-like functional equation $f(x+g(x)y)=f(x)+f(g(0)y)-f(0)$ | There are other solutions which is similar to your last one. They are: $f(x)=\ln(x+\frac{b}{a})$, and $g(x)=ax+b$ for $a\neq 0$, $b$ constants. I think this answers your question.
And for the general solution, if one assumes both functions are differentiable. There would be no other solutions except what we listed. T... | 1 | https://mathoverflow.net/users/99556 | 251997 | 114,285 |
https://mathoverflow.net/questions/251988 | 6 | This question is motivated by the "interesting tidbit" in Hamkins' response here: <https://mathoverflow.net/a/99025/10671>, in which he demonstrates that, after Cohen forcing, there is a perfect set consisting entirely of mutually generic Cohen reals. In particular, this demonstrates that there is a perfect set in the ... | https://mathoverflow.net/users/10671 | Is there a perfect set of ground model reals in the Cohen extension? | A very nice question!
The answer is no, there cannot be a perfect set in $V[c]$
consisting entirely of ground-model reals.
Suppose towards contradiction that there is a such a set. So this
set consists of the paths through a certain perfect tree $T\subset
2^{<\omega}$ in $V[c]$.
Note, as an easy first case, that ... | 11 | https://mathoverflow.net/users/1946 | 252001 | 114,287 |
https://mathoverflow.net/questions/252006 | 10 | Duke, and Linnik before him under a restrictive condition, proved that the set of closed geodesics of a given length $L$ is equidistributed on the modular surface as $L \to \infty$. This is a theorem of a deep arithmetic significance, which is powered by Siegel's ineffective theorem (but is semi-effective in the sense ... | https://mathoverflow.net/users/26522 | Refined equidistribution for the periodic trajectories of Anosov flows? | Since you mention the related question of grouping together geodesics of lengths in the interval $[L,L+1]$, let me point out that this case is covered by the 1972 result of Bowen that I mentioned in the comments: [[R. Bowen, "Periodic orbits for hyperbolic flows", *Amer. J. Math* **94** (1972), 1-30]](http://www.ams.o... | 4 | https://mathoverflow.net/users/5701 | 252007 | 114,289 |
https://mathoverflow.net/questions/252003 | 5 | A few years ago, Schechtman showed that $\ell\_p(\ell\_q)$ fails to admit a greedy basis whenever $1\leq p\neq q<\infty$. This furnishes an example of a Banach space with an unconditional basis but not a greedy one. However, I am also interested in the following:
**Question 1.** Does there exist a Banach space $X$ ad... | https://mathoverflow.net/users/73784 | Banach space with an unconditional basis but not a quasi-greedy one? | In their Absolutely Summing Operators paper, Lindenstrauss and Pelczynski gave positive answers to questions 2, 3, and 4. You can find a proof on p. 29 of the Basic Concepts article Joram and I wrote for the Handbook of the Geometry of Banach Spaces. (This proof uses Khintchine's inequality rather than Grothendieck's i... | 4 | https://mathoverflow.net/users/2554 | 252010 | 114,291 |
https://mathoverflow.net/questions/251466 | 0 | I'm interested if there is the explicit forms of basis functions in $L^2(S^n), n\geq 3$.
For $n=1, n=2$ basis functions are well known: $\{e^{ik\phi}\}\_{k\in\mathbb{Z}}$, $\{p^{|m|}\_n(\cos \gamma) e^{im\phi} | (\gamma,\phi)\in S^2, n \geq 0, m = \overline{-n,n}\}$. Also the following fact is known:
\begin{equation... | https://mathoverflow.net/users/94631 | Basis on the sphere in multidimensions | Ivan Izmestiev answered my question, see comments below.
In addition, the question above inspired me to find the explicit result for such integrals (they arise in the framework of generalized Radon transforms):
<https://hal.archives-ouvertes.fr/hal-01415990>
| 1 | https://mathoverflow.net/users/94631 | 252017 | 114,296 |
https://mathoverflow.net/questions/252000 | 14 | Let $G\subset \mathrm{SL}\_2(\mathbb R)$ be a subgroup such that $\mathrm{SL}\_2(\mathbb Z)\subset G$.
What are the possible groups such that $\mathrm{SL}\_2(\mathbb Z)\subset G$ is of finite index? Is $G=\mathrm{SL}\_2(\mathbb Z)$ the only possibility?
What if we replace $\mathrm{SL}\_2$ by $\mathrm{Sp}\_{2n}$?
... | https://mathoverflow.net/users/99622 | Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup | Even for $Sp\_{2g}$ the only possibility is $Sp\_{2g}(\mathbb{Z})$. To see this, suppose $\Gamma \subset Sp\_{2g}(\mathbb{R})$ is a subgroup containing $Sp\_{2g}(\mathbb{Z})$ as a finite index subgroup. Suppose we prove that $\Gamma $ consists of rational symplectic matrices. The fact that $Sp\_{2g}(\mathbb{Z}\_p)$ is ... | 17 | https://mathoverflow.net/users/23291 | 252021 | 114,297 |
https://mathoverflow.net/questions/252033 | 2 | If $X\subset \mathbb{P}^{n}$ is a cubic hypersurface that is not normal, what's the easiest way to see that the nonnormal locus is a linear subspace of dimension $n-2$?
As for a reference, there is a paper that classifies the nonnormal cubic hypersurfaces that gives a proof by expressing the hypersurface as a cone o... | https://mathoverflow.net/users/16356 | Nonnormal locus of cubic hypersurface | Singularities of a hypersurface, supported in codimension 2, are normal. So, the question is, what are the $(n-2)$-dimensional components of the singular locus of $X$. Denote the union of these components by $Z$.
Note that the secant variety $Sec(Z)$ of $Z$ is contained in $X$ (indeed, any secant of $Z$ intersects $X... | 4 | https://mathoverflow.net/users/4428 | 252036 | 114,302 |
https://mathoverflow.net/questions/251968 | 7 | Given a sequence $a\_1, a\_2,\dots,a\_n$, define the two sequences
$$l\_i=\max\_{1 \leq j < i, a\_j \geq a\_i} j$$
or $0$ if it does not exist; and
$$r\_i=\min\_{i < j \leq n, a\_j > a\_i} j$$
or $n+1$ if does not exist.
I want to find a permutation of $1, 2, ..., n$ so that
$$\sum\_{1 \leq i \leq n} \min(i-l\_i,... | https://mathoverflow.net/users/94928 | How to find a permutation of [n] so that $\sum\{\min(i-l[i],r[i]-i)\}$ is maximized? | Denote the maximum of your sum by $f(n)$, agree also that $f(0)=0$. If we fix $k$ such that $a\_k=n$, then the maximal possible value is $\min(k,n+1-k)+f(k-1)+f(n-k)$. So, we get a recursive formula $$f(n)=\max\_{1\leqslant k\leqslant (n+1)/2} k+f(k-1)+f(n-k).$$
Now we may forget about permutations and study this recur... | 4 | https://mathoverflow.net/users/4312 | 252041 | 114,304 |
https://mathoverflow.net/questions/251502 | 2 |
>
> Let $f(x,y)$ be a non-negative function with $x,y \in \mathbb R^3$ that satisfies
> $$
> I\_1(f) := \iint\_{\mathbb R^3 \times \mathbb R^3 } f(x,y) \, dx \ dy < \infty
> $$
> and
> $$
> I\_2(f) := \iint\_{\mathbb R^3 \times \mathbb R^3 } |y|^2 f(x,y) \, dx \ dy < \infty.
> $$
> Also, define the sequence by
> ... | https://mathoverflow.net/users/89456 | Bounding a function with second moments | First let me remark that in this case one can make an explicit computation using the explicit values of first and second moments of Gaussian.
Then note that the $x$ variable does not play any role, as when we integrate by parts, we get the normalised Gaussian which just give $1$. What caught my attention was this $\Ga... | 1 | https://mathoverflow.net/users/56191 | 252042 | 114,305 |
https://mathoverflow.net/questions/252009 | 10 | What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}\_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes?
Is there, perhaps, even an intrinstic characterization of those locally ringed toposes which are equivalent to $(\mathbf{Sh}(X)... | https://mathoverflow.net/users/98306 | Properties of the petit Zariski topos | Unfortunately I don't know an interesting intrinsically formulated sufficient criterion for a locally ringed topos to be the little Zariski topos of a scheme. This is an extremely interesting question!
There are necessary conditions, for instance (formulated in the internal language of the topos):
>
> For any ele... | 11 | https://mathoverflow.net/users/31233 | 252046 | 114,306 |
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