parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/375921 | 5 | Suppose we are given a cube and we add a pair of crossing edges inside each of its faces. It is clear that this drawing has 6 crossings. My question is whether such a graph has crossing number 6? How to prove or disprove it?
(there is a new question below)
[What is the crossing number of dodecahedron with a copy of... | https://mathoverflow.net/users/148974 | What is the crossing number of cube with a pair of crossing edges inside each face | Yes, the crossing number is $6$.
In general if a graph with $V$ vertices and $E$ edges
can be drawn without crossings then $E \leq 3V-6$,
by a familiar application of Euler's formula $V-E+F = 2$:
there are $F = E-V+2$ faces; but each edge separates $2$ faces
while each face has at least $3$ sides; so
$$
2E \geq 3F = ... | 8 | https://mathoverflow.net/users/14830 | 375924 | 156,821 |
https://mathoverflow.net/questions/375930 | 7 | Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma\_1 < \sigma\_2$ satisfy $\sigma\_1 \sigma\_2=1$.
Does there exist an area-preserving diffeomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ whose singular values are constant $\sigma\_1 , \sigma\_2$?
---
An immedi... | https://mathoverflow.net/users/46290 | A diffeomorphism of the torus with constant singular values | There are no non-affine examples with $f$ smooth (or even $C^3$). This follows from the fact that such an $f:\mathbb{T}^2\to\mathbb{T}^2$ would lift to a smooth (local) diffeomorphism $F:\mathbb{R}^2\to\mathbb{R}^2$ with constant singular values, and the argument I gave in my answer to [this question](https://mathoverf... | 7 | https://mathoverflow.net/users/13972 | 375931 | 156,822 |
https://mathoverflow.net/questions/375932 | 3 | This question is motivated by two examples of locally nilpotent groups which I came across (see below).
**Question:** Given an infinite solvable and locally nilpotent group $G$, does $G$ have an infinite centre?
The question concerns groups which are not finitely generated (otherwise the answer is well-known [yes])... | https://mathoverflow.net/users/18974 | Centre of solvable locally nilpotent groups | No.
For a scalar (= unital associative commutative) ring $R$, consider $V=V(R)=R[X]$, the polynomial ring, and $q$ the operator $X^n\mapsto X^{n-1}$, $X\mapsto 0$. Let $V\_n=V\_n(R)$ be the $R$-submodule of degree $\le n$ polynomials. Then $q$ stabilizes $V\_n$ and is nilpotent on $V\_n$. Hence $1+q$ is invertible on... | 3 | https://mathoverflow.net/users/14094 | 375945 | 156,823 |
https://mathoverflow.net/questions/375906 | 3 | Is there an accessible proof for the following fact?
>
> If $A=C\_0(X)$ with $X$ locally compact Hausdorff and $B$ is a
> $C^\ast$-algebra then $M(A\otimes B)$ is the set of bounded strictly
> continuous functions $X \to M(B)$.
>
>
>
Denote the set of bounded strictly continuous functions by $C\_b^s (X, M(B))$... | https://mathoverflow.net/users/nan | Identifying the multiplier $C^*$-algebra $M(C_0(X) \otimes B)$ | Indeed, Blackadar appears to have made a mistake here. But the reference he gives is good, and seems to give both a correct statement, and a proof:
Akemann, Charles A.; Pedersen, Gert K.; Tomiyama, Jun
Multipliers of C∗-algebras.
J. Functional Analysis 13 (1973), 277–301.
[MR470685](https://mathscinet.ams.org/maths... | 2 | https://mathoverflow.net/users/406 | 375947 | 156,824 |
https://mathoverflow.net/questions/375934 | 4 | Suppose $G$ is an infinite graph with vertices labelled as $v\_{i,j}$ such that $i,j$ are positive integers (so we assume that there are denumerable many vertices of the graph). For any two positive integers $i,j$ we join the vertex $v\_{i,j}$ with all the vertices $v\_{k,i+j}$ for every $k \in \mathbb{N}.$
Is the ch... | https://mathoverflow.net/users/109471 | If $G$ is an infinite graph where $v_{i,j}$ is joined with $v_{k,i+j}$ for all $k,i,j$ then is the chromatic number of G always infinite? | The chromatic number is indeed infinite.
Assume that there is a proper coloring in finitely many colors. Denote by $S\_i$ the set of colors of the vertices having the form $v\_{k,i}$ (for some $k$). There are two equal sets, say $S\_i$ and $S\_j$ with $i<j$. Then the color of $v\_{j-i,i}$ lies in that set, and theref... | 5 | https://mathoverflow.net/users/17581 | 375957 | 156,826 |
https://mathoverflow.net/questions/375928 | 2 | Suppose $X$ is a smooth scheme, $E=O\_X^{\oplus n}$ and $\varphi\in SL\_n(E)$, i.e. $\varphi$ has trivial determinant and is an isomorphism. Is the morphism
$$\mathbb{P}(\varphi)^\*:CH^{\bullet}(\mathbb{P}(E))\longrightarrow CH^{\bullet}(\mathbb{P}(E))$$
equal to the identity map?
| https://mathoverflow.net/users/149491 | Invariance of Chow groups of projective bundles under automorphisms of bundles | This is true for any smooth variety $X$ (i.e. separated scheme of finite type over a field) and any vector bundle $E$ if and only if $\mathbb{P}(\varphi)^\*(\mathcal{O}\_{\mathbb{P}(E)}(1))=\mathcal{O}\_{\mathbb{P}(E)}(1)$.
For one direction, note that if this equality doesn't hold, then the automorphism induces a mo... | 1 | https://mathoverflow.net/users/65919 | 375961 | 156,827 |
https://mathoverflow.net/questions/375953 | 2 | Given a von Neumann algebra $M$, let
$$
S(M) = \{u\in M: uu^\*u=u\}
$$
be the set of partial isometries in $M$. Given $u,v\in S(M)$, it is well known that $uv \in S(M)$, provided $u^\*u$ commutes
with $vv^\*$.
Now suppose that $N$ is another von Neumann algebra and that
$$
\varphi :S(M)\to S(N)
$$
is a bijective ... | https://mathoverflow.net/users/97532 | Von Neumann algebras with isomorphic sets of partial isometries | No, consider the map $\phi: u \mapsto u^\*$ from the partial isometries in $M$ to the partial isometries in its opposite algebra $M^{op}$. One easily checks that it has the desired properties, but it cannot extend to a linear map because already e.g. $\phi(iI) = -iI \neq iI = i\phi(I)$.
If $M$ is not isomorphic to it... | 1 | https://mathoverflow.net/users/23141 | 375965 | 156,829 |
https://mathoverflow.net/questions/375968 | 14 | Call an $n$-vector $v$ in $\mathbb{Z}^n$ cool when it has only entries 0 or 1 and the ones appear in only one block. Thus there are $n(n+1)/2$ such vectors. For $n=3$ they are:
[ <[ 1, 0, 0 ]>, <[ 1, 1, 0 ]>, <[ 0, 1, 0 ]>, <[ 1, 1, 1 ]>, <[ 0, 1, 1 ]>, <[ 0, 0, 1 ]> ].
Let $X\_n$ be the set of cool $n$-vectors. Ca... | https://mathoverflow.net/users/61949 | A canonical bijection from linear independent vectors to parking functions | They are in canonical bijection with the spanning trees of the complete graph $K\_{n+1}$ (for which the bijections with parking functions are well known).
Indeed, let $K\_{n+1}$ be the complete graph on the ground set $\{0,1,\ldots,n\}$. Denote $f\_0=0$ and consider $n$ linearly independent vectors $f\_1,\ldots,f\_n$... | 19 | https://mathoverflow.net/users/4312 | 375972 | 156,833 |
https://mathoverflow.net/questions/375969 | 2 | Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}\_{X|G}$, how to compute cohomology of this vector bundle and related vector bundle, i.e, what is $H^k(X,\mathcal{N}\_{X|G}\otimes\mathcal{O}\_X(2))$, $H^k(X,\math... | https://mathoverflow.net/users/41650 | Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold | $\mathcal{N}\_{X|G}$ is in your case $F|\_X$, where $F=O\_G(1)^2 \oplus O\_G(2))$. In order to compute the first two spaces, you can simply use the Koszul complex for X $$ 0 \to det(F^{\vee}) \to \wedge^2 F^{\vee} \to F^{\vee} \to O\_G \to O\_X \to 0,$$ twisted with $F$ (or $F(2)$ in the second case).
You can use Bor... | 6 | https://mathoverflow.net/users/52811 | 375985 | 156,835 |
https://mathoverflow.net/questions/375980 | -1 | $$\frac{d}{dx}\int\_{0}^{1}|log\_2(1+t)-(t+x)|\,dt=0$$
Is this solvable at all?
| https://mathoverflow.net/users/168420 | How to solve this equation for x? | The points where $\log\_2(1+t)-(t+x)$ changes sign can be expressed in terms of the Lambert W function, and then the integral can be evaluated explicitly
and differentiated. According to Maple, you want to solve
$$ \left( -2\,{\rm W}\_{-1} \left(-{2}^{x-1}\ln \left( 2 \right) \right)-2
\right) {\rm W} \left(-{2}^{x-1}... | 1 | https://mathoverflow.net/users/13650 | 375988 | 156,836 |
https://mathoverflow.net/questions/375977 | 9 | Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\mathcal X\_+$ be the set of all elements of $\mathcal X$ with a nonnegative version (a closed convex cone), and $\mathcal Y... | https://mathoverflow.net/users/145424 | Closedness of linear image of positive L1 functions | Take $\mathcal X = L^1(\Omega,\mu)$ where $\Omega = \{1,2,3,\dots\}$ and measure $\mu(\{k\}) = p\_k$ with $p\_k > 0$,
$\sum p\_k = 1$. The norm in $\mathcal X$ is
$$
\|f\|\_{\mathcal X} = \sum\_k |f(k)|p\_k .
$$
Let $\mathcal Y = \mathbb R^2$ with norm
$$
\|(x,y)\|\_{\mathcal Y} = \frac{1}{2}|x|+\frac{1}{2}|y|.
$$
Thus... | 10 | https://mathoverflow.net/users/454 | 375994 | 156,838 |
https://mathoverflow.net/questions/375987 | -1 | Landau's 4th problem asks if $\,L\_n=n^2 + 1\,$ is prime for infinitely many $n \in \Bbb{N}$. This obviously occurs only when $n$ is even.
Let's consider composite positive integers represented by
$$\,(n+1)^4+4=(n^2+1)\cdot((n+2)^2+1)=L\_n\cdot L\_{n+2}$$
If the number of Landau's primes was finite, let it be $L\_M... | https://mathoverflow.net/users/150698 | About Landau's 4th problem | Suppose $k^4+4=L\_{k-1}L\_{k+1}$ has fewer than four prime divisors. Since the factors are relatively prime, this implies that one of them is a prime power. But since $L\_{k\pm 1}$ is one larger than a square, from Mihailescu's theorem (more specifically its special case when one of the powers is a square) we get that ... | 7 | https://mathoverflow.net/users/30186 | 375995 | 156,839 |
https://mathoverflow.net/questions/375998 | 15 | This question was inspired by the following:
<https://math.stackexchange.com/questions/3882691/lfloor-xn-rfloor-lfloor-yn-rfloor-is-a-perfect-square>
Is there a real nonintegral $x>1$ s.t. $\lfloor x^n \rfloor$ is square integer for all positive integers $n$? I am asking because the question is interesting in and o... | https://mathoverflow.net/users/122188 | Is there a real nonintegral number $x >1$ such that $\lfloor x^n \rfloor$ is a square integer for all $n \in \mathbb{N}$? | There is no such number. Suppose $\alpha>1$ is a real number such that $\lfloor \alpha^n \rfloor$ is a square for all $n\in {\Bbb N}$. Put $\beta=\sqrt{\alpha}$.
Now for each $n$ we have
$$
m^2 + 1 > \alpha^n \ge m^2
$$
for some integer $m$, so that taking square-roots
$$
m + \frac{1}{2m} > \beta^n \ge m.
$$
In ot... | 37 | https://mathoverflow.net/users/38624 | 376002 | 156,842 |
https://mathoverflow.net/questions/375946 | 11 | $\DeclareMathOperator{\SO}{\operatorname{SO}}\DeclareMathOperator{\SU}{\operatorname{SU}}$**Background:** In the quantum field theory literature people commonly consider "expansions" of $\SO(n)$ or $\SU(n)$ field theories in $1/n$ (see [this wikipedia article](https://en.wikipedia.org/wiki/1/N_expansion)). While this e... | https://mathoverflow.net/users/109191 | Representation theory of $\operatorname{SO}(n)$ for large $n$ | In this answer I'll focus on the representation theory of $SO(n)$ as $n \to \infty$ (rather than the group theory). Strictly speaking, I want to discuss $O(n)$ rather than $SO(n)$, but I hope that it's good enough for your purposes. There are two approaches that I would like to discuss. Both of them involve a category ... | 17 | https://mathoverflow.net/users/159272 | 376008 | 156,844 |
https://mathoverflow.net/questions/375432 | 3 | Let $X \subseteq \mathbb{R}^n$ be a closed $d$-dimensional regular set (i.e. for any $x \in X$ and $0<r< \text{diam(X)}>$, $\mathscr{H}^d(B(x,r)) \sim r^d$ ) which has the property that for any $y \in \mathbb{R}^n \setminus X$, the closest points from $y$ to $X$ are of the form $z\_{\pm}=(y\_1,...,y\_{n-1},y\_{n} \pm r... | https://mathoverflow.net/users/168019 | Distance function and geometry of the set | Since Pietro Majer definitely was right about the structure of the set but hasn't supplied a proof let me jump in with an elementary one. I think the problem is to specific to find a reference, but it might have occurred to others in some context unbeknownst to me.
**Lemma:** If the closest point in $X$ is always dir... | 0 | https://mathoverflow.net/users/51695 | 376023 | 156,847 |
https://mathoverflow.net/questions/376028 | 0 | Given $ \mathbb{E}X^2<\infty $, how can I show that if two $\sigma$-algebras $\mathscr{G}\_1\subset \mathscr{G}\_2$, then $\mathbb{E}[Var(X|\mathscr{G}\_2)]\leq \mathbb{E}[Var(X|\mathscr{G}\_1)]$ ?
I have noticed that $\mathbb{E}[Var(X|\mathscr{G})] = \mathbb{E}X^2-\mathbb{E}[\mathbb{E}^2(X|\mathscr{G})]$ and was think... | https://mathoverflow.net/users/168447 | Why does the dispersion of X about its conditional mean decreases as the σ−algebra grows? | $\newcommand\G{\mathscr G}$For $j=1,2$, let $E\_j$ and $V\_j$ denote, respectively, the conditional expectation and the conditional variance given $\G\_j$. Given that $\G\_1\subset\G\_2$, you want to show that $E V\_2 X\le E V\_1 X$. You have already established that $E V\_j X=EX^2-EY\_j^2$ for $j=1,2$, where $Y\_j:=E\... | 0 | https://mathoverflow.net/users/36721 | 376045 | 156,850 |
https://mathoverflow.net/questions/375974 | 3 | Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\Delta$ in $S$. It is well-known that the Betti numbers (graded or otherwise) appearing in the free resolution of $I$ may... | https://mathoverflow.net/users/94968 | Are there characteristic-dependent Betti numbers in characteristic not equal to two? | Consider the presentation complex of $\mathbb{Z}/3\mathbb{Z}=\langle x|x^3=1 \rangle$, consisting of a full triangle with all three edges identified in the same orientation. It has a triangulation with 9 vertices and facets $abd,acd,abf,acf,abe,ace,def,bcg,ceg,deg,bdg,bch,cfh,efh,beh,bci,cdi,dfi,bfi$.
The projective ... | 2 | https://mathoverflow.net/users/160416 | 376047 | 156,851 |
https://mathoverflow.net/questions/350282 | 21 | I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum\_{i,j=1}^da\_{ij}(x)D\_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a non-real eigenvalue in $L^2(\Omega)$.
Under the above assumptions, the operator $L$ has a discrete spectrum and the ... | https://mathoverflow.net/users/150653 | Non real eigenvalues for elliptic equations | Here is a construction. It elaborates from perturbation analysis of eigenvalues. However it starts from the situation of a non-simple eigenvalue.
So, let me start with the standard self-adjoint $L\_0=-\Delta$. I assume that the Dirichlet problem admits an eigenvalue $\lambda\_0$ of multiplicity $2$ exactly. I denote ... | 19 | https://mathoverflow.net/users/8799 | 376056 | 156,852 |
https://mathoverflow.net/questions/376058 | 4 | \*\*Disclaimer:\*\*I posted the following question on [MSE](https://math.stackexchange.com/posts/3900271/edit), but since there were no answers. I'm migrating it here.
Let $Homeo\_0(\mathbb{R}^n)$ ($Homeo\_c(\mathbb{R}^n)$) be the space of all (compactly-supported) orientation-preserving homeomorphisms on $\mathbb{R}... | https://mathoverflow.net/users/36886 | Density of compactly-supported homeomorphisms | I think this is true. It suffices to prove the
**Lemma**. Given an orientation-preserving homeomorphism $h$ of $\mathbb{R}^n$ there is a compactly-supported homeomorphsim $h\_1$ which agrees with $h$ on the unit ball.
By conjugating by dilations one can find $h\_r$'s that agree with $h$ on the ball of radius $r$, a... | 3 | https://mathoverflow.net/users/318 | 376062 | 156,854 |
https://mathoverflow.net/questions/376055 | 2 | ### Definitions
Let $\mathcal{T}$ be a triangulated category with translation functor $\Sigma$.
* **Generator**: An object $T$ of $\mathcal{T}$ is a generator of $\mathcal{T}$ if $\mathcal{T}(\Sigma^n T,A)=0$ for all $n$ implies $A=0$.
* **$T$-ghost**: A map $f:A\to B$ in $\mathcal{T}$ is called $T$-ghost if the re... | https://mathoverflow.net/users/118028 | $T$ is a generator of $\mathcal{T}$ iff. the ideal of $T$-ghost maps is contained in the Jacobson radical of $\mathcal{T}$ | (i) $\implies$ (ii): If $f \colon A \to B$ is a $T$-ghost and $g \colon B \to A$ is arbitrary, then $gf$ is a $T$-ghost and therefore $1\_A - gf$ induces the identity map $\mathsf{Hom}(\Sigma^n T,A) \to \mathsf{Hom}(\Sigma^n T,A)$ for all $n$. If $C$ is the cone of $1\_A -gf$ then the long exact sequence for Hom shows ... | 2 | https://mathoverflow.net/users/1310 | 376085 | 156,857 |
https://mathoverflow.net/questions/376105 | 5 | What are examples of non-homeomorphic connected $T\_2$-spaces $(X\_i,\tau\_i)$ for $i=1,2$ such that the posets $(\tau\_1, \subseteq)$ and $(\tau\_2,\subseteq)$ are order-isomorphic?
| https://mathoverflow.net/users/8628 | Non-homeomorphic connected $T_2$-spaces with isomorphic topology poset | There aren't any. Hausdorff spaces are [sober spaces](https://ncatlab.org/nlab/show/sober+topological+space). If $X, Y$ are sober, then every frame map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite meets and arbitrary joins, arises from a uniquely determined c... | 17 | https://mathoverflow.net/users/2926 | 376107 | 156,864 |
https://mathoverflow.net/questions/376093 | 3 | $\DeclareMathOperator\ord{ord}$This is related to a question in the MO post, [Does there exist a prime $p$ such that $\left|\frac{\mathrm{ord\_{p}}(a)}{\mathrm{ord\_{p}}(b)}-c\right|<\gamma$ for some small constant $\gamma$?](https://mathoverflow.net/q/376040/160943)
Let $p$ be a prime and $\ord\_{p}(a)$ be the least... | https://mathoverflow.net/users/160943 | Does there exist a prime $p$ such that $ \frac{\operatorname{ord}_{p}(a)}{\operatorname{ord}_{p}(b)}>1?$ | This question might be difficult, but heuristically it should be true. Essentially the same heuristic that yields the Artin primitive root conjecture, there should be for any such $a$ and $b$ be infinitely many primes $p$ where $a$ is a primitive root mod $p$, and $b$ is not.
Another heuristic approach is that if $a$... | 2 | https://mathoverflow.net/users/127690 | 376115 | 156,869 |
https://mathoverflow.net/questions/376036 | 2 | Let $X$ be a riemannian surface. Suppose $f:X\to X$ is a Teihmuller map with respect to a quadratic differential $q$ on $X$. This means that, if $q=dz^2$ in local coordinates in a neighborhood of nonzero point of $q$, then $f=Kx+\frac1Ky$ for $z=x+iy$ and for a constant $K$ which does not depend on local coordinates.
... | https://mathoverflow.net/users/76500 | Why a Teichmuller map is not a pseudo-anosov? | The map $f$ is a Teichmuller map with respect to **a pair** of quadratic differentials $q\_1,q\_1$ on $X$ --- initial differential and terminal differential. More precisely, $f$ maps the horisontal/vertical foliation of $q\_1$ to the horisontal/vertical foliation of $q\_2$.
To verify that $f$ is pseudo-anosov we need... | 0 | https://mathoverflow.net/users/76500 | 376116 | 156,870 |
https://mathoverflow.net/questions/376087 | 6 | To fix the ideas, let's work on the flat periodic torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$.
My question is the following.
>
> Does there exist an infinite dimensional Banach (sub-)algebra $A \subset \mathscr{C}^\infty(\mathbf{T}^d)$ stable by derivation ?
>
>
>
I am almost certain that the answer is n... | https://mathoverflow.net/users/27767 | Banach algebra of smooth functions | I am not 100% sure that I am correctly interpreting your hypotheses, so let me write something which is more cumbersome but which hopefully applies to the situation you have in mind.
FACT 1. Let $A$ be a commutative Banach algebra and let $D:A\to A$ be a continuous derivation, i.e. a bounded linear map satisfying $D(... | 2 | https://mathoverflow.net/users/763 | 376131 | 156,875 |
https://mathoverflow.net/questions/376098 | 3 | During research I came to the following sequence:
Let $\lambda>1$
and define $n\_{k+1}=\text{IntergerPart}[\lambda\cdot n\_k]$ where we assume that $n\_0$ is sufficently large integer, so that the sequence $n\_k$ is strictly increasing. Finally let $x\_k=\text{FractionalPart}[\lambda\cdot n\_k]$.
**Question:** Are ... | https://mathoverflow.net/users/47862 | Properties of a certain sequence | Claim 1 is false. Let $\phi = \frac{1+\sqrt{5}}{2}$, $\overline{\phi} = \frac{1-\sqrt{5}}{2}$, $\lambda = \phi^{2}$ and $n\_{0} = 1$.
I claim that $n\_{k} = F\_{2k+1}$, the $(2k+1)$st Fibonacci number for $k \geq 1$.
By strong induction and Binet's formula $F\_{k} = \frac{1}{\sqrt{5}}\left(\phi^{k} - \overline{\phi}^... | 3 | https://mathoverflow.net/users/48142 | 376133 | 156,876 |
https://mathoverflow.net/questions/376124 | 7 | I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry.
As far as I understand, two statements should be true, but I cannot find a reference:
Notation:
$B^n$ is the $n$-dimensional disk (with boundary).
Given PL manifolds $M^m,N^n $ with $m... | https://mathoverflow.net/users/128408 | Singularities of PL embedding of surface in a contractible 4-manifold | One reference for these matters is Rourke-Sanderson "Introduction to piecewise-linear topology". In particular, if $S\subset M$ is a simplical submanifold, then Corollary 4.2 in this book implies that $S$ is locally knotted in $M$ if and only if for each interior vertex of $S$ the pair $(\text{link of $x$ in $S$}, \tex... | 6 | https://mathoverflow.net/users/1573 | 376141 | 156,879 |
https://mathoverflow.net/questions/376143 | 6 | Let $S\_g^n$ be a surface of genus g with n boundaries and let $Mod(S\_g^n)$ be its mapping class group.
We will also denote by $S\_{g,m}^n$ a surface of genus g with n boundaries and m punctures.
The classical Birman exact sequence is
$$ 1 \to \pi\_1(UT(S\_g^{n-1}) \to Mod(S\_g^n) \to Mod(S\_g^{n-1}) \to 1$$
whe... | https://mathoverflow.net/users/150711 | Generalized Birman exact sequence for surfaces with boundaries | It's almost right, your guess.
The easiest description is geometric, generalizing Birman's description $\pi\_1(UT(S\_g^{n-1}))$. You look instead at the configuration space parametrizing $k$ distinct points in $S\_g^{n-k}$, together with unit tangent vectors at each of these $k$ points. So this is an open subspace of... | 6 | https://mathoverflow.net/users/1310 | 376148 | 156,881 |
https://mathoverflow.net/questions/376129 | 5 | Let $E$, $M$ be smooth finite dimensional, Hausdorff and second-countable manifolds. Let $\pi:E \longrightarrow M$ be a surjective submersion.
$\pi$ is locally trivial if $\forall p\in M$, $\exists U \ni p$ open neighborhood such that there is a diffeomorphism:
$$ \phi:\pi^{-1}(U)\longrightarrow U \times \pi^{-1}(p)$... | https://mathoverflow.net/users/157138 | What are the sufficient and necessary conditions for surjective submersions to be locally trivial | A connection for a (surjective) submersion $\pi\colon E\to M$ is a complementary subbundle $\mathcal H E\subset TE$ of the vertical bundle $\mathcal VE=ker (D\pi).$ A connection locally defines a parallel transport in $E$ along curves $\gamma\colon I\to M$, by lifting to a horizontal curve, i.e., $$\hat\gamma\colon \ti... | 3 | https://mathoverflow.net/users/4572 | 376152 | 156,882 |
https://mathoverflow.net/questions/375933 | 2 | Let $S$ be a set of $n \gg 1$ points lying on the interval $[0,1]$. Given a point $p\in[0,1]$, let $S\_p\subseteq S\times S$ be the set formed by all pairs of points $(x,y)$ with $x,y\in S$, such that either $\max(x,y)\le p$ or $\min(x,y)\ge p$. Finally let $d(S\_p)=\frac{1}{|S\_p|}\sum\_{(x,y)\in S\_p} |x-y|$ be the a... | https://mathoverflow.net/users/115803 | Probabilistic combinatorial optimization problem on the distances between pairs of points in $[0,1]$ | Here is an approach that gives a lower bound, that I expect to be tight.
The first step is to observe that if $\mu$ is a non-atomic
probability distribution on $[0,1]$,
$(X\_i)\_{i=1}^n$ are iid and $\mu$ distributed, and $L\_n=n^{-1} \sum\_{i=1}^n \delta\_{X\_i}$ the associated empirical measure, then
$$ m\_n\geq E\_\... | 1 | https://mathoverflow.net/users/35520 | 376157 | 156,885 |
https://mathoverflow.net/questions/376139 | 1 | As a lover of number theory, the condition $f(xy)=f(x)f(y)$ in group morphisms always reminds me of completely multiplicative functions. Since the natural numbers $\mathbb{N}$ do not form a group under multiplication, we will work instead in the non-zero rational numbers $\mathbb{Q}^\*$ where the value of the multiplic... | https://mathoverflow.net/users/159298 | Does viewing multiplicative functions as morphisms from $\mathbb{Q}^*\to\mathbb{C}^*$ have any consequnces? | Actually number theorists more-or-less do the *exact opposite* of what you suggest.
Namely, as explained in the comments, a multiplicative function is determined by its values on the primes. This suggest more that a multiplicative function should be thought of as a local object, rather than a global object.
So what... | 3 | https://mathoverflow.net/users/5101 | 376163 | 156,887 |
https://mathoverflow.net/questions/376177 | 0 | Let $A$ be an $n\times n$ random matrix with i.i.d. $N(0,\sigma)$ entries, for some $\sigma>0$ and let $x\in \mathbb{R}^n$. A direct computation shows that $Ax \sim N(0,\sigma x^{\top}x)$.
I would like to sample from $N(0,\sigma x^{\top}x)$ however, the trouble is that if $n$ is large then this storing the matrix $x^... | https://mathoverflow.net/users/36886 | Algorithm for economically sampling method for Gaussian matrix product | You can sample from the product $Ax$ in the following way:
Sample a row of $A$ and multiply by $x$. To save memory, forget the row. Sample another row of $A$ and multiply by $x$. To save memory, forget the row. And so on.
An even faster way of sampling from $a^T x$ where $a$ is a row of $A$ is to simply sample from... | 1 | https://mathoverflow.net/users/75761 | 376182 | 156,891 |
https://mathoverflow.net/questions/252716 | 5 | **Question:** Is the ring of real-analytic functions on $\mathbb{C}\mathbb{P}^n$ (real valued)
a Noetherian ring?
References or counterexamples are welcome.
I know that the ring of germs of holomorphic functions on $\mathbb{C}^n$ is Noetherian, but the ring of holomorphic functions on $\mathbb{C}^n$ is not ! I also... | https://mathoverflow.net/users/47862 | Do real analytic functions on $\mathbb{C}\mathbb{P}^n$ form a Noetherian ring? | Since I found the answer to my question I will write it here, hoping that it will be helpful to someone else.
**Answer:** The ring of global real analytic functions on a compact real analytic Stein manifold is noetherian due to Theorem I.9. [J.Frisch.: *Points de platitude d'un morphisme d'espaces analytiques complex... | 8 | https://mathoverflow.net/users/47862 | 376185 | 156,892 |
https://mathoverflow.net/questions/376171 | 4 | $\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that [extends the addition on $\omega$ in a natural way](http://people.dm.unipi.it/dinasso/ultracombinatorics/strauss.pdf). (The same... | https://mathoverflow.net/users/8628 | Addition and Rudin-Keisler ordering in $\beta \omega$ | For Question 1: First, there are idempotent ultrafilters, so some ultrafilters satisfy 1 in a very strong form. But 1 does not hold in general. The reason is that the semigroup $\beta\omega-\omega$ has subsemigroups $G$ that are groups of cardinality $2^{\mathfrak c}$ where $\mathfrak c$ is the cardinal of the continuu... | 9 | https://mathoverflow.net/users/6794 | 376195 | 156,896 |
https://mathoverflow.net/questions/376043 | 2 | What cardinal is the limit of this fundamental sequence?
{The first Mahlo cardinal, the first 1-Mahlo cardinal, the first hyper-Mahlo cardinal, the first hyper-hyper-Mahlo cardinal, the first hyper-hyper-hyper-Mahlo cardinal, ...}
where the definition of a $\alpha$-Mahlo cardinal is as follows:
If we define a fun... | https://mathoverflow.net/users/168460 | Limit of Mahlo cardinals | There is some problems with definitions here.
1. **The definition you use** of $\alpha+1$-Mahlo cardinals is a cardinal $\kappa$ such that $\kappa$ is the $\kappa$th $\alpha$-Mahlo cardinal.
**This is not the standard definition**. Usually, a cardinal is said to be $\alpha+1$-Mahlo if $\{\beta\lt\kappa|\beta\text{ ... | 2 | https://mathoverflow.net/users/141402 | 376212 | 156,903 |
https://mathoverflow.net/questions/376187 | 5 | A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e.,
if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t\_0)=y(t\_0)$ then
$$\cos (Tx(t\_0),Ty(t\_0))= \cos (x(t\_0),y(t\_0))= \frac{x'(t\_0)\cdot y'(t\_0)}{|x'(t\_0)|| y'(t\_0)|}.$$
A typical example of conformal ma... | https://mathoverflow.net/users/112207 | Looking for a reference on conformal mapping on $\Bbb R^n$ | My two cents: a proof for $n=3$ is given explicitly by Dubrovin, Fomenko and Novikov in [1], §15.2 pp. 138-142. The authors explain also how to extend their proof to the case $n>3$ and leave the details as an exercise.
**An addendum**
The answer by @Piotr Hajlasz triggered my curiosity and pushed me to go a little ... | 7 | https://mathoverflow.net/users/113756 | 376213 | 156,904 |
https://mathoverflow.net/questions/376210 | 5 | Let $\Lambda$ be finite dimensional algebra over a field $k$. The (left) finitistic dimension of a finite dimensional algebra is defined as
$$\operatorname{findim}(\Lambda)=\sup\{\operatorname{pd}M | M \in \operatorname{mod}\Lambda,\operatorname{pd}M < \infty\}$$
where $\operatorname{mod}\Lambda$ is the category of... | https://mathoverflow.net/users/157483 | Is there a finite dimensional algebra with left finitistic dimension different from its right finitistic dimension? | The left and right finitistic dimension can take any pair of values. This is shown in Example 2.3 of
*Green, Edward L.; Kirkman, Ellen; Kuzmanovich, James*, [**Finitistic dimensions of finite dimensional monomial algebras**](http://dx.doi.org/10.1016/0021-8693(91)90062-D), J. Algebra 136, No. 1, 37-50 (1991). [ZBL072... | 7 | https://mathoverflow.net/users/22989 | 376214 | 156,905 |
https://mathoverflow.net/questions/375131 | 11 | Let $\sigma$ be a permutation of $[k]=\{1,2, \dots , k\}$. Consider all the ordered triples $(\pi, s\_{1},s\_{2})$, such that $\pi$ is a permutation of length $2k-1$ that is a union of its two subsequences $s\_{1}$ and $s\_{2}$, each of which is of length $k$ and is order-isomorphic to $\sigma$.
Example:
$\sigma = ... | https://mathoverflow.net/users/85939 | The number of ways to merge a permutation with itself | By @Max Alexeyev's solution above $N\_{2k-1}^{\sigma}=tr(M\_{k}(P\_{\sigma}M\_{k}P\_{\sigma}^{-1}))$.
The eigenvalues and eigenvectors of $M\_k$ are given here: [Result attribution for eigenvalues of a matrix of Pascal-type](https://mathoverflow.net/questions/278400/result-attribution-for-eigenvalues-of-a-matrix-of-p... | 8 | https://mathoverflow.net/users/48831 | 376218 | 156,906 |
https://mathoverflow.net/questions/376199 | 4 | Assume $(A\_i)\_{i \in I}$ is a family of locally convex topological
vector spaces which are all moreover assumed to be Banach spaces.
We assume moreover that $(A\_i)\_{i \in I}$ has additional
structure of an inductive system, that is $I$ is a directed pre-ordered set and
for all $i < j$ there are compatible continu... | https://mathoverflow.net/users/108274 | Compatibility of inductive and projective limits with dualization in functional analysis | Here are some remarks on the inductive limit case:
1. The dual of the inductive limit is ALWAYS identifiable (as a vector space) with the projective limit of the duals. This is just the universal property of the inductive limit;
2. The question of whether they are isomorphic as locally convex spaces is a much more de... | 5 | https://mathoverflow.net/users/131781 | 376224 | 156,910 |
https://mathoverflow.net/questions/376247 | 2 | Let $M^3$ be a closed, connected and oriented smooth $3$-manifold, and fix an integer $g \geq 1$. Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and orientable embedded minimal surfaces of genus $g$ in $M$ is compact? If not, is it true for a generic set of metrics of positi... | https://mathoverflow.net/users/85934 | Space of embedded minimal surfaces of fixed genus in a generic $3$-manifold | The set of embedded minimal surfaces $\Sigma$ with $\textrm{genus}(\Sigma) \leq g\_0$ and $\textrm{area}(\Sigma)\leq A\_0$ is compact for a generic metric (this follows from White's version of the Choi--Schoen compactness theorem <https://mathscinet.ams.org/mathscinet-getitem?mr=880951>, see also Appendix A: <https://m... | 4 | https://mathoverflow.net/users/1540 | 376261 | 156,920 |
https://mathoverflow.net/questions/376113 | 14 | I'm following Arone's lectures on Goodwillie calculus from Munster 2015. There he left an exercise:
>
> Find $\partial\_kF$ for $F: \text{Top}\_\* \to \text{Sp}$ given by $F(X) = \Sigma^\infty X^{\wedge n}/\Sigma\_n$ where $X^{\wedge n} = X \wedge \cdots \wedge X$ and $\Sigma\_n$ is the symmetric group, which acts ... | https://mathoverflow.net/users/137622 | Goodwillie derivatives of $X \mapsto \Sigma^\infty X^{\wedge n}/\Sigma_n$ | Your solution is correct as far as it goes. Let $\rho^{n-1}\cong {\mathbb R}^{n-1}$ denote the reduced standard representation of $\Sigma\_n$. One way to define $\rho^{n-1}$ explicitly is to say that it is the orthogonal complement of the diagonal in $\mathbb R^n$. Let's write it out:
$$
\rho^{n-1}=\{(x\_1, \ldots, x\_... | 14 | https://mathoverflow.net/users/6668 | 376278 | 156,924 |
https://mathoverflow.net/questions/375276 | 9 | Let $M \in \mathbb{R}^{k\times k}$ positive definite with $\operatorname{tr} M = m$, where $m$ is an integer such that $m \geq k$. I have found a way (using [this](https://mathoverflow.net/a/280203/89544) answer) to decompose $M = AA^t$ with $A \in \mathbb{R}^{k \times m}$ such that $A = (a\_1, \dots, a\_m), a\_i \in \... | https://mathoverflow.net/users/89544 | $M = AA^t$ where $A$ has unit norm columns | This decomposition is equivalent to write $M$ as a sum of rank one orthogonal projection $$ M = \sum\_{i=1}^m a\_i a\_i^\* $$
with $\|a\_i\|=1$. Indeed for any $x$ we have $$(Mx)\_{s} = \sum\_{i\leq m,t\leq k} A\_{si}A^T\_{it}x\_t = \sum\_{i\leq m} (a\_i)\_s \langle a\_i,x\rangle
$$
Remark that in form it is easy to se... | 2 | https://mathoverflow.net/users/99045 | 376279 | 156,925 |
https://mathoverflow.net/questions/375793 | 5 | Let $X$ be an $n$-dimensional smooth algebraic variety, and let $Y$ be a compactification with $E=Y\setminus X$ simple normal crossings. There is the natural quotient map
$$\Omega\_Y^n(\log E)\to \mathcal{H}^n(\Omega\_Y^\bullet(\log E))\to 0.$$
I want to understand the image of this map on global sections. If I am not ... | https://mathoverflow.net/users/64302 | Is there a way to describe the image of the $n$-fold residue map from $H^0(Y,\Omega_Y^n(\log E))$? | Your guess in the last sentence is not quite right, but there is a natural higher-dimensional generalization of the statement. What happens is that you have a "boundary map"
$$ \bigoplus\_{p\_i} \mathbb C \to \bigoplus\_{C\_j} \mathbb C$$
where the points $p\_i$ are all the $n$-fold intersections of boundary components... | 4 | https://mathoverflow.net/users/1310 | 376280 | 156,926 |
https://mathoverflow.net/questions/376221 | 9 | Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal transformation taking $R$ to $R'$ (these transformations are not required to do anything nice to the rest of the arrangement). Is $\mat... | https://mathoverflow.net/users/33089 | Hyperplane arrangements whose regions all have the same shape | This is a known open problem (for isometric regions), which, as far as I know, is still not settled.
The dimension 3 case was proved affirmatively in <https://arxiv.org/abs/1501.05991>, where also some history of the question is outlined. I am not aware of any progress since.
| 11 | https://mathoverflow.net/users/21291 | 376282 | 156,927 |
https://mathoverflow.net/questions/376292 | 4 | It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B\_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, let $B\_{\alpha}$ be the collection of complements of elements and unions of countable sequences of $B\_{\beta}$. For $\... | https://mathoverflow.net/users/138274 | Is every element of $\omega_1$ the rank of some Borel set? | In $\mathsf{ZFC}$ the Borel hierarchy of an uncountable Polish space $X$ has length exactly $\omega\_1$, meaning that $\mathbf{\Sigma}^0\_\xi(X)\neq\mathbf{\Pi}^0\_\xi(X)$ for all $\xi<\omega\_1$.
The standard way of proving this is through the construction of so called $\mathcal C$-universal sets for $\mathbf{\Sigma... | 17 | https://mathoverflow.net/users/49381 | 376296 | 156,932 |
https://mathoverflow.net/questions/376293 | 2 | Let $\{U\_i\}\_{i=1}^2$ be an open cover of $S^1$, with $U\_i\cong \mathbb{R}$ (for example, $U\_1$ is the lower arc of the circle and $U\_2$ is the upper part). Let $\iota\_i:U\_i\hookrightarrow S^1$ be the cannonical inclusions and let $\iota\_{i\star}$ be the push-forwards on the function spaces $C(\mathbb{R},U\_i)$... | https://mathoverflow.net/users/36886 | Density of functions into the circle glueing | Maybe I don't understand your question. But isn't $\iota\_{i\*}[C(\mathbb R,U\_i)]$ the set of continuous functions with values in $U\_i$? Why should the set of functions with values in either of the subsets be dense? For example, $f(x)=e^{ix}$ is not in the closure of the union.
| 2 | https://mathoverflow.net/users/21051 | 376300 | 156,933 |
https://mathoverflow.net/questions/376271 | 0 | Recall that $(y\_{n})\_{n}$ is a convex block subsequence of a sequence $(x\_{n})\_{n}$ in a Banach space $X$ provided that there exists a strictly increasing sequence of positive integers $(k\_{n})\_{n}$ so that $y\_{n}\in \textrm{co}(x\_{i})\_{i=k\_{n-1}+1}^{k\_{n}}$ for every $n$ ($k\_{0}=0$).
Let us recall that a... | https://mathoverflow.net/users/41619 | A characterization of Grothendieck spaces via convex block subsequences | Let $X$ be such that every weak\*-null sequence $(x\_n^\*)$ in $X^\*$ admits a convex block sequence which is norm-null. Since subsequences of weak\*-null sequences are again weak\*-null, by the converse to Mazur it follows that $(x\_n^\*)$ is weakly null.
| 1 | https://mathoverflow.net/users/73784 | 376308 | 156,935 |
https://mathoverflow.net/questions/376228 | 11 | Every finite field of characteristic $2$ ist given by $\mathbb{F}\_2[x]/P(x)$ for some irreducible polynomial $P\in \mathbb{F}\_2[x]$.
For small degree, a simple algorithm gives a way to find $P$. Is there a way to give explicitely the polynomial $P$ for large degrees? I would be interested in a formula like
$$x^n+a\... | https://mathoverflow.net/users/23758 | Explicit large finite fields in characteristic $2$ | For $n=3^k$, the polynomial $p=x^{2n}+x^n+1$ is irreducible over $\mathbb{F}\_2$.
**Proof:** By Rabin's irreducibilty test, it suffices to check that $p|x^{2^{2n}}-x$ and $\gcd(p,x^{2^{2n/3}}-x)=1$.
Note that the order of $x$ mod $p$ is $3n=3^{k+1}$. Hence, since $3^{k+1}|4^{n}-1$ by lifting-the-exponent lemma, we ... | 5 | https://mathoverflow.net/users/160416 | 376309 | 156,936 |
https://mathoverflow.net/questions/376266 | 6 | Let $G$ be a finite group. Two irreducible complex representations $V,V'$ of $G$ are called *dual* to each other if $V \otimes V'$ admits a trivial component, i.e. $\hom\_G(V \otimes V',V\_0)$ is positive dimensional (thus one-dimensional) with $V\_0$ the trivial representation. Then the representation $V'$ is denoted ... | https://mathoverflow.net/users/34538 | A property forcing the Frobenius-Schur indicator to be positive | Here is a more general statement, see also Lemma 1.2 in [1].
>
> **Lemma:** Let $Z$ be a self-dual $kG$-module which admits a non-degenerate $G$-invariant symmetric (alternating) bilinear form $b$. Suppose that $W$ is a self-dual irreducible $kG$-module. If $W$ occurs in $Z$ as a composition factor of odd multiplic... | 9 | https://mathoverflow.net/users/10146 | 376313 | 156,937 |
https://mathoverflow.net/questions/376239 | 15 | I came up with the following conjecture:
$$
\sum\_{n \ge 0} z^n \sum\_{\mu \vdash n} \frac{ t^{\sum l}q^{\sum a}}{\prod (q^a - t^{l+1})(t^l - q^{a+1})} = \exp\left(\sum\_{n \ge 1} \frac{z^n}{n(q^n-1)(t^n-1)}\right)
$$
where each unlabeled sum and product is over the cells of the diagram of the partition, and $l$ and $a... | https://mathoverflow.net/users/19088 | A formula for this generating function that is similar to the $qt$-Catalan numbers | This identity can be proved using the results in Garsia and Haiman's paper ["A Remarkable q,t-Catalan Sequence and q-Lagrange Inversion"](https://link.springer.com/article/10.1023/A:1022476211638). In particular theorem 3.10(e) gives
$$\sum\_{\mu \vdash n} \frac{ t^{\sum l}q^{\sum a}}{\prod (q^a - t^{l+1})(t^l - q^{a+1... | 7 | https://mathoverflow.net/users/2384 | 376326 | 156,940 |
https://mathoverflow.net/questions/376288 | 12 | If we define $$f(n) = \Bigl\{ n \sum\_{k=2}^{n-1} \frac{1}{k}\Bigr\}$$
is it true that $f(n) \ne f(m)$ whenever $n \ne m, \forall m,n \in \Bbb{N}$ (where the curly braces denote the fractional part)?
I wanted to explore coming up with a kind of "global residue" concept for a number, based on all the numbers less th... | https://mathoverflow.net/users/155247 | Is $\Bigl\{ n \sum_{k=2}^{n-1} \frac{1}{k}\Bigr\}$ unique $\forall n \in \Bbb{N}, n>1$ | It follows from Bertrand's postulate that $mH\_m - nH\_n$ is an integer only in the case $n=m$. Suppose $n<m$, and assume that $m$ is reasonably large below (say $\ge 20$).
Let $p$ denote the largest prime below $m$, so that by Bertrand's postulate $p>m/2$. The only term with $p$ in the denominator in $mH\_m$ is $m/p... | 13 | https://mathoverflow.net/users/38624 | 376331 | 156,942 |
https://mathoverflow.net/questions/376299 | 7 | Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, then the image of the moment map is a hexagon with vertices are image of $M^T$ by $\mu $.
My questions are:
$1.$ Wha... | https://mathoverflow.net/users/168620 | Question about an example in symplectic geometry | I'll preserve your notation: $M$ is the coadjoint orbit of a regular semisimple element $X \in \mathfrak t^\*$ (which you seem to also call $x$). I also assume we're working in characteristic $0$, or at least not $3$.
The orbit $M$ is neither contained in, nor contains, $\mathfrak t^\*$. Rather, a conjugate of $X$ li... | 5 | https://mathoverflow.net/users/2383 | 376333 | 156,943 |
https://mathoverflow.net/questions/376343 | 9 | Let $E\_1, \dots, E\_N$ be independent events, each of probability $p$, where $p$ is very close to $0$. Let $A\_N = \frac{1}{N} ( 1\_{E\_1} + \dots + 1\_{E\_N} )$ be the proportion of the events $E\_i$ that occur. We expect $A\_N$ to be tightly concentrated around its mean $p$.
Suppose we want to estimate something l... | https://mathoverflow.net/users/166445 | Concentration inequalities for very rare events on a multiplicative scale | Let $n:=N$. Let us show that for all natural $n$ and all $p\in(0,1)$
$$P(A\_n>\sqrt p)\le\frac{\sqrt p+p}{1+p},\tag{1}$$
so that $P(A\_n>\sqrt p)\to0$ whenever $p\downarrow0$.
Consider first the case when $n\ge1/\sqrt p$, so that $1/n\le\sqrt p$.
In view of [Cantelli's inequality](https://en.wikipedia.org/wiki/Cantel... | 7 | https://mathoverflow.net/users/36721 | 376348 | 156,946 |
https://mathoverflow.net/questions/376183 | 5 | Let $B^4$ be the closed unit ball in $\mathbb{C}^2$ and $J$ an almost complex structure sufficiently closed to the standard complex structure on $\mathbb{C}^2$ in the $C^0$-topology. Let $u \colon S \rightarrow B^4$ be a smooth $J$-holomorphic curve transverse to $\partial B^4$. Can $u$ extend to a $\tilde{J}$-holomorp... | https://mathoverflow.net/users/41200 | Extension of a holomorphic curve in $B^4$ to one in $\mathbb{C}P^2$ | Yes, you can.
The "boundary" $\partial B \cap {\rm Im}(u)$ of $u$ is a link $K$ that is transverse to the contact structure $JTS^3\cap TS^3$, which is isotopic to the standard contact structure on $S^3$. John Etnyre and I [proved](http://arXiv.org/abs/2001.08978) that every transverse link in $(S^3,\xi\_{st})$ can be... | 3 | https://mathoverflow.net/users/13119 | 376355 | 156,948 |
https://mathoverflow.net/questions/376330 | 3 | Let $X$ be a smooth variety and let $Y\subset X$ be a closed smooth subvariety. We have the local cohomology $H^i\_Y(X)$, which becomes a MHS via the mixed cone construction, as explained in Section 5.5 from "Mixed hodge structures" by Peters and Steenbrink.
One also has the Thom isomorphism $H\_Y^i(X)\cong H^{i-c}(Y... | https://mathoverflow.net/users/64302 | How does the MHS on $H_Y^i(X)$ behave with respect to the Thom isomorphism? | The Thom isomorphism is an isomorphism of mixed Hodge structures up to a Tate twist:
$$ H^k\_Y(X,\mathbf Q) \cong H^{k-2c}(Y,\mathbf Q) \otimes \mathbf Q(-c).$$
I don't know what's the canonical reference for this fact - how you prove it will in any case depend on how you choose to define the MHS on the left hand side ... | 4 | https://mathoverflow.net/users/1310 | 376358 | 156,950 |
https://mathoverflow.net/questions/376366 | 6 | I apologize in advance if this question is too elementary for MO. I am new to the field of algebraic geometry.
I am dealing with a (real) algebraic variety $V$ of (Krull) dimension $n$. I keep reading that $V$ can be decomposed into differentiable manifolds of dimensions $0, 1, ..., n$.
While this seems plausible I... | https://mathoverflow.net/users/111720 | Decomposition of real algebraic varieties into manifolds | I believe this follows from results in Chapter 9 of *Real Algebraic Geometry* by Bochnak, Coste, and Roy. In particular, Proposition 9.1.8 implies the strata are [Nash manifolds](https://en.wikipedia.org/wiki/Nash_functions#Nash_manifolds), satisfying certain "niceness" conditions.
| 7 | https://mathoverflow.net/users/124323 | 376370 | 156,954 |
https://mathoverflow.net/questions/376369 | 12 | I roll a fair die with $n>1$ sides until the most recent roll is smaller than the previous one. Let $E\_n$ be the expected number of rolls. Do we have $\lim\_{n\to\infty} E\_n < \infty$? If not, what about $\lim\_{n\to\infty} E\_n/n$ and $\lim\_{n\to\infty} E\_n/\log(n)$?
| https://mathoverflow.net/users/8628 | Throwing a fair die until most recent roll is smaller than previous one | By doing casework on the second to last roll $m$, one has $$E\_n = \sum\_{k=2}^\infty k p\_k^{(n)},$$ where $$p\_k^{(n)} = \sum\_{m=1}^n n^{-(k-1)}{m+k-3 \choose k-2}\frac{m-1}{n}.$$ Note, for any fixed $k$, $$\lim\_{n \to \infty} p\_k^{(n)} = \frac{k-1}{k!}.$$ Therefore, by an easy dominated convergence argument, one ... | 17 | https://mathoverflow.net/users/129185 | 376372 | 156,955 |
https://mathoverflow.net/questions/376362 | 1 | In Terence Tao's paper [Exploring the toolkit of Jean Bourgain](https://arxiv.org/pdf/2009.06736.pdf) is stated:
Theorem 3.1 (Furstenberg–Katznelson–Weiss theorem, qualitative version). Let $A\subset\Bbb R^2$ be a measurable set whose upper density $$δ∶=\limsup\_{R→∞}\frac{|A∩\mathrm B(0,R)|}{|\mathrm B(0,R)|}$$is po... | https://mathoverflow.net/users/7458 | Why does $l_0$ appear in this statement of the Furstenberg–Katznelson–Weiss theorem? | I looked up [the paper](https://mathscinet.ams.org/mathscinet-getitem?mr=1083601) in which Furstenberg-Katznelson-Weiss proved their result. The correct statement has $|x-y| = \ell$ instead of $|x-y| \geq \ell$.
| 2 | https://mathoverflow.net/users/37327 | 376374 | 156,956 |
https://mathoverflow.net/questions/376373 | 2 | I'm looking at the Gamma space construction in the paper "Gamma spaces and information" by Matilde Marcolli. For a pointed set $X$ one takes the category $P(X)$ of pointed subsets of $X$ and inclusions, a category $\mathcal{C}$ with coproducts and a zero, and then looks at the functor category $\Sigma\_\mathcal{C}(X)$ ... | https://mathoverflow.net/users/168659 | Functor category with only isomorphisms in the Gamma space construction | Unfortunately I have not read Marcolli's paper, but it sounds like you are describing a construction that goes back to Segal in ``Categories and cohomology theories''.
I think the short answer to your question is people take the category of isomorphisms because in this way one tends to get the most interesting homoto... | 2 | https://mathoverflow.net/users/6668 | 376376 | 156,958 |
https://mathoverflow.net/questions/376352 | 2 | Given a connected Lie group $G$ with corresponding Lie algebra $\mathfrak{g}$, the adjoint representation/action $\mathrm{Ad} : G \to \mathrm{GL}(\mathfrak{g})$ induces a Lie group homomorphism. It's well-known that $\ker{\mathrm{Ad}} = Z(G)$, and that if $G$ is semisimple, then $\mathrm{Ad}(G)$ is the identity compone... | https://mathoverflow.net/users/146831 | Embedding of the adjoint group into $\mathrm{GL}(\mathfrak{g})$ | I think the answer is no.
Fix an irrational number $r\_0\in \mathbf{R}\setminus\mathbf{Q}$.
Let $G = \mathbf{R}\ltimes \mathbf{C}^2$ be the semidirect product, where $\mathbf{R}$ acts on $\mathbf{C}^2$ by
$$
t\cdot(z\_1, z\_2) = (e^{2\pi t\sqrt{-1}}z\_1, e^{2\pi r\_0t\sqrt{-1}}z\_2).
$$
Then $\varphi:G / Z(G)\to \ma... | 3 | https://mathoverflow.net/users/38052 | 376379 | 156,960 |
https://mathoverflow.net/questions/376356 | -1 | Let $x\_n$ be a non-negative valued sequence and suppose that the following hold:
* $\lim\limits\_{n\to\infty} x\_n =0$
* There exists some polynomial function $p$ of degree at-least $1$ such that:
$$
\|x\_n\| \leq p(n^{-1});
$$
(not that there is no assumption that $p(0)=0$!)
* $x\_n\neq 0$ for any $n>0$.
Can we d... | https://mathoverflow.net/users/36886 | Inferring polynomial rate of convergence from polynomial bound | The answer is no (assuming that $\|x\|:=x$ for real $x\ge0$). Indeed, as noted by mathworker21, your condition on the existence of $p$ is always satisfied, by choosing, e.g., $p(t)=c+t$ with $c:=\sup\_n x\_n$.
On the other hand, your condition on the existence of $q$ will not hold e.g. if $x\_n=1/\sqrt n$ for all nat... | 1 | https://mathoverflow.net/users/36721 | 376387 | 156,963 |
https://mathoverflow.net/questions/376380 | 3 | I recently came across the notion of **comprehension** for a fibration of categories $p:E\to B$ that subsumes the axiom of separation and the grothendieck construction for fibred categories. A nice definition can be found in the [n-lab article on the axiom of separation](https://ncatlab.org/nlab/show/axiom+of+separatio... | https://mathoverflow.net/users/1261 | Reference Request: Comprehension for multicategories | In [Comprehensive factorisation systems](https://arxiv.org/abs/1710.09438), Berger and Kaufmann establish a correspondence between certain orthogonal factorisation systems (the motivating example being the discrete fibration/final functor OFS on $\mathbf{Cat}$) and consistent comprehension schemes. In Section 2, they e... | 4 | https://mathoverflow.net/users/152679 | 376396 | 156,965 |
https://mathoverflow.net/questions/376336 | 2 | I am trying to better understand a condition that appears in Theorem 1 of [this paper](https://arxiv.org/abs/1510.07921).
Let $K$ be a convex and compact subset of a locally convex tvs. The condition is:
>
> $K$ embeds linearly into a strictly convex dual Banach space endowed with its weak\* topology. Call this t... | https://mathoverflow.net/users/96899 | Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space? | The space $M$ of finitely additive probability measures on $\omega\_1,$ with the weak topology defined by the seminorms $\mu\mapsto|\mu(S)|$ for $S\subseteq\omega\_1,$ does not admit a strictly convex lower semicontinuous function. This answers your question because of the easy direction of Theorem 1.1 of that paper - ... | 4 | https://mathoverflow.net/users/164965 | 376412 | 156,972 |
https://mathoverflow.net/questions/376405 | 1 | This question is regarding the convolution theorem. So far, I have only used it as a tool to solve some recurrence relations. A nice example that I am familiar with is the renewal formula for finding first passage probabilities $-$ consider a continuous time Markov process on a discrete state space $\mathcal{S}$. We de... | https://mathoverflow.net/users/168672 | Convolution in state space of a Markov process? | $\newcommand\S{\mathcal S}\newcommand\V{\mathcal V}\newcommand\tC{\tilde C}\newcommand\tD{\tilde D}$
The answer is: in general, no. Indeed, suppose the contrary: that you have an imbedding $f$ of your state space $\S$ into a vector space $\V$ such that for
$$Q(D',t|D):=
\left\{
\begin{aligned}
P(f^{-1}(D'),t|f^{-1}(D))... | 1 | https://mathoverflow.net/users/36721 | 376418 | 156,974 |
https://mathoverflow.net/questions/376408 | 2 | Is Birkhoff's pointwise/individual ergodic theorem for $L\_\infty.$ Clearly, it is true if the measure space is finite? What about the measure space not finite?
| https://mathoverflow.net/users/136860 | Is Birkhoff's ergodic theorem true for $L_\infty$? | Here is a counter-example to the question I think you're asking. Let $X$ be the space of two-sided 0-1 valued sequences. Let $x$ the sequence with $x\_n=0$ for all $n\le 0$; $x\_n=1$ if $2^k<n\le 2^{k+1}$ for any even $k\ge 0$ and $x\_n=0$ if $2^k<n\le 2^{k+1}$ for any odd $k\ge 0$. Then form an infinite measure by $\m... | 3 | https://mathoverflow.net/users/11054 | 376426 | 156,979 |
https://mathoverflow.net/questions/376431 | 3 | Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, set $${\bf a}+{\bf b} = \big\{ N\subseteq \omega:\{x \in \omega:\{y\in\omega: x+y\in N\}\in {\bf b}\}\in {\bf a}\big\}.... | https://mathoverflow.net/users/8628 | "Completion property" in $(\beta\omega,+)$ | No. Kunen proved that there are many weak P-points in $\beta\omega-\omega$, which means they are nonprincipal ultrafilters that are not in the closure of any countable set of other nonprincipal ultrafilters. In particular, a weak P-point is never a sum of nonprincipal ultrafilters because $\mathbf a+\mathbf b$ is in th... | 9 | https://mathoverflow.net/users/6794 | 376432 | 156,980 |
https://mathoverflow.net/questions/376425 | 3 | Consider the equation
$$\partial\_t u(t,x) = -\partial\_{xxx} u(t,x)$$
for $x \in \mathbb R.$
It is well-known that in general we have
$$\Vert u(t) \Vert\_{L^{\infty}} \le C t^{-1/3} \Vert u\_0 \Vert\_{L^1}.$$
I now read in some lecture notes, without proper argument, that for $u\_0$ such that $\operatorname{su... | https://mathoverflow.net/users/108483 | Improving dispersive estimates for linear KdV | Let $\chi$ be a smooth function with support on $[-3,-1/3] \cup [1/3,3]$ and equals 1 on $[-2,-1/2] \cup [1/2,2]$.
You can write
$$ u(t,x) = \int e^{it \xi^3}e^{ix\xi} \chi(\xi) \hat{u}\_0(\xi) ~d\xi $$
if $u\_0$ has Fourier support as you demanded.
If you write
$$ V(t,x) = \int e^{it\xi^3}e^{ix\xi} \chi(\xi) ~d\xi$$... | 2 | https://mathoverflow.net/users/3948 | 376433 | 156,981 |
https://mathoverflow.net/questions/263097 | 3 | Suppose $U$ is an open subset of $\mathbb{R}^n$, and $f:U\to \mathbb{R}$. When $f$ is $C^2$ we know that the mixed partial derivatives are symmetric, i.e.
$\partial\_i\partial\_jf= \partial\_j\partial\_if.$
But as it is famous the continuity of the 2nd order partial derivatives is not necessary for this to happen. Fo... | https://mathoverflow.net/users/35800 | Symmetry of higher order mixed partial derivatives under weaker assumptions | For the accurate statement and rigorous proof see Theorem 3 of the following paper <https://www.mdpi.com/2227-7390/8/11/1946/htm>
| 1 | https://mathoverflow.net/users/48826 | 376440 | 156,987 |
https://mathoverflow.net/questions/376454 | 6 | The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
| https://mathoverflow.net/users/168671 | Do you know which is the minimal local ring that is not isomorphic to its opposite? | I learned this example from MO-user Johannes Hahn:
The algebra is $A=K<x,y>/(x^3,y\*x,y^2,x^2\*y)$ over a field $K$ with 2 elements.
Then $A$ as an $A$-module as 20 submodules, but $A^{op}$ as an $A^{op}$-module has 16 submodules.
Thus $A$ and $A^{op}$ are not isomorphic.
This also gives an example where $A$ and $A^{... | 6 | https://mathoverflow.net/users/61949 | 376461 | 156,993 |
https://mathoverflow.net/questions/376463 | -2 | Is this series absolutely converges
$$
\sum\_{n=1}^{\infty}C\_nU\_n
$$
when
$\sum\_{n=1}^{\infty}C\_n$ is absolutely convergent series and
$\sum\_{n=1}^{\infty}U\_n$ is conditionally convergent series?
| https://mathoverflow.net/users/163626 | When multiply absolutely convergent seires and conditionally convergent series | I don't think this could be a considered a research level question: perhaps, this could be defined a ~~tricky~~ simple exercise. However, the answer is yes as I am going to show below
If $\sum\_{n=1}^\infty C\_n$ is absolutely convergent, then we have that its [total variation as a sequence](https://en.wikipedia.org/... | 1 | https://mathoverflow.net/users/113756 | 376468 | 156,995 |
https://mathoverflow.net/questions/376446 | 18 | Recall that the ring of Gaussian integers is
$$\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}.$$
Clearly
$$(a+bi)^2=a^2-b^2+2abi\ \ \mbox{and}\ \ (a+bi)^4=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i.$$
**Question.** Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?
**Evidence.** Via Mathematica I... | https://mathoverflow.net/users/124654 | Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$? | Yes, it is true that $\{ x^{4} + y^{2} + z^{2} : x, y, z \in \mathbb{Z}[i] \} = \{ a + 2bi : a, b \in \mathbb{Z} \}$. Indeed, one can even take $x$ to be either $0$ or $1$ in all cases. Because $y^{2}+z^{2} = (y+iz)(y-iz)$ is reducible, this is analogous to the statement that every integer can be written in the form $x... | 27 | https://mathoverflow.net/users/48142 | 376471 | 156,997 |
https://mathoverflow.net/questions/376448 | 2 | **EDIT**:
1. *First edit after an interesting answer*.
2. *$(S,\mathcal{T}\_1)$ and $(S,\mathcal{T}\_2)$ are homotopy equivalent to the same Quillen cofibrant space*.
Let $S$ be a set with two topologies $\mathcal{T}\_1$ and $\mathcal{T}\_2$ such that the identity map induces a continuous map $(S,\mathcal{T}\_1)\to... | https://mathoverflow.net/users/24563 | Continuous bijection between two homotopy equivalent $\Delta$-generated spaces | Let $X = S^1$ and $X^\delta$ its discretisation. Then the identity map $\iota: X^\delta \to X$ is a continuous bijection of $\Delta$-generated spaces, although not a homotopy equivalence. But the map of unreduced cones $$C(\iota): C(X^\delta) \to C(X)$$ is a homotopy equivalence and a continuous bijection, and both spa... | 6 | https://mathoverflow.net/users/168706 | 376474 | 156,998 |
https://mathoverflow.net/questions/376272 | 3 | As I'm just a layperson I don't understand the technicalities involved, but does the paper *New Large Cardinal Axioms and the Ultimate-L Program*, by Rupert McCallum (arXiv:[1812.03837](https://arxiv.org/abs/1812.03837)) prove the inconsistency of a Reinhardt cardinal? If so can someone explain how given the results of... | https://mathoverflow.net/users/168572 | Inconsistency of Reinhardt cardinals in ZF+DC | Before explaining the results, there are some facts that should be noted:
1. Arxiv papers are not peer-certified scientific papers. It is very possible, and in fact very common, for false theorems to appear on Arxiv. They are endless papers detailing proofs of the inconsistency of $ZFC$ for instance.
2. Rupert McCall... | 5 | https://mathoverflow.net/users/141402 | 376484 | 157,000 |
https://mathoverflow.net/questions/376191 | 14 | The [Lee-Yang circle theorem](https://freedommathdance.blogspot.com/2018/05/a-theorem-of-lee-yang.html) states that if $\left( a\_{ij} \right)$ is a Hermitian square $n \times n$ matrix whose entries are in the closed unit disc, then the polynomial $$ P\left(Z \right) = \sum\_{S\subseteq\left[n\right]}\left(\prod\_{i\i... | https://mathoverflow.net/users/103908 | Converse of the Lee-Yang circle theorem for polynomials with unitary roots | It looks to be enough to require this for $\lambda=1$.
Indeed, assume that the polynomial $Q(Z)=\sum\_{k=0}^n \beta\_k Z^k$, $\beta\_0=\beta\_n=1$, has all roots on the unit circle. Denote the roots by $-\theta\_i^{-1}$, $i=1,\ldots,n$. Then $Q(Z)=\prod (1+\theta\_i Z)$ and $\prod\_i \theta\_i=1$.
There exists an H... | 8 | https://mathoverflow.net/users/4312 | 376485 | 157,001 |
https://mathoverflow.net/questions/376494 | 14 | I know that the identity
$$
s\_\mu = \sum\_{\mu-\lambda \text{ is a horizontal strip}} \;\sum\_{\alpha\vdash|\lambda|} \frac{\chi^\lambda\_\alpha}{z\_\alpha} \prod\_i(p\_i-1)^{a\_i}
$$
holds.
Here $\alpha=1^{a\_1}2^{a\_2}\dotsb$ in exponential notation; in other words, $a\_i$ is the number of times that $i$ occurs in $... | https://mathoverflow.net/users/9672 | Do you know an elegant proof for this expression for a Schur function? | I would like to suggest an interpretation using super symmetric functions. These are symmetric functions that are symmetric in two sets of variables $\{x\_i\}$ and $\{y\_j\}$ separately. They satisfy the property that setting a single $x\_i$ variable to equal $z$ and a single $y\_j$ variable equal to $z$ gives a polyno... | 12 | https://mathoverflow.net/users/159272 | 376495 | 157,004 |
https://mathoverflow.net/questions/376493 | 10 | Let $(S, \Sigma)$ be a measurable space. Let $f: S \rightarrow \mathbb{R}$ be function and let $\mathcal{B}(\mathbb{R})$ be Borel $\sigma$-algebra on $\mathbb{R}$. Let $G(f)$ be the graph of $f$, that means
$ G(f)=\{(x,f(x)) \in S\times \mathbb{R} : x \in S\} $. We write $ \Sigma \otimes \mathcal{B}(\mathbb{R})$ to ind... | https://mathoverflow.net/users/76004 | Is there an example of a non measurable function with a measurable graph? | See Srivastava, *A Course on Borel Sets*, in the section "Solovay's Coding of Borel Sets", beginning at "We now proceed to give an example of a function with domain coanalytic whose graph is Borel and that is not Borel measurable." [Google books preview](https://books.google.co.uk/books?id=46LyCAAAQBAJ&pg=PA152&dq=doma... | 11 | https://mathoverflow.net/users/164965 | 376501 | 157,006 |
https://mathoverflow.net/questions/376490 | 4 | Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, *On topological cyclic homology*, arXiv:[1707.01799](https://arxiv.org/abs/1707.01799) (on p147).
---
I am still lost. But from Maxime's helpful comment and replies, let me list out my concerns - which are listed as (X),(Y),... | https://mathoverflow.net/users/97321 | Computing homotopy colimit of a space with free $S^1$-action | Note that in their context, $C$ has an action of $B\mathbb Z$, not of $\mathbb Z$ ! (Otherwise $C/B\mathbb Z$ wouldn't make sense)
This amounts essentially to a self natural transformation of the identity functor
For the first claim and the commutative square, this is true as geometric realization is a left adjoint $... | 4 | https://mathoverflow.net/users/102343 | 376504 | 157,007 |
https://mathoverflow.net/questions/376517 | 1 | Let $U$ be a smooth variety, and $U\hookrightarrow X$ an smooth compactification with snc boundary $D=X\setminus U$. Suppose that $\omega\in H^0(U,\Omega^n\_U)$ is global algebraic $n$-form on $U$. It defines a class in $H^n(U,\mathbb{C})=\mathbb{H}^n(X,\Omega\_X^\bullet(\log D))$.
The form $\omega$ extends to a mero... | https://mathoverflow.net/users/64302 | Is every $\omega\in H^0(U,\Omega_U^n)$ representable in $H^n(U,\mathbb{C})$ by an element from $H^0(X,\Omega_X^n(\log D))$? | This is not true. Take for $X$ an elliptic curve, for $D$ a point $p\in X$. The restriction map $H^1(X,\mathbb{C})\rightarrow H^1(U,\mathbb{C})$ is an isomorphism, and $H^0(X,\Omega ^1\_X(\log D))=H^0(X,\Omega ^1\_X)$. There is a form $\tilde{\omega } $ with a pole of order 2 at $p$; its restriction $\omega $ to $U$ is... | 4 | https://mathoverflow.net/users/40297 | 376519 | 157,010 |
https://mathoverflow.net/questions/376332 | 5 | I know the basic idea behind the renormalization group approach as it is used in mathematical physics to study both QFT and statistical mechanics. However, I have trouble understanding how can one recover the information one was trying to obtain using this technique. Let me elaborate.
Although there is no such thing ... | https://mathoverflow.net/users/150264 | How can one recover/obtain information from the renormalization group procedure? | 1. The limiting function $V^\ast$ is such that any further convolutions of $e^{-V^\ast}$ with $\mu$ return $e^{-V^\ast}$, so $Z^\ast(\phi)=e^{-V^\ast(\phi)}$.
2. To obtain critical properties, you need the correlator $K(x,x')=\langle\phi(x)\phi(x')\rangle$. The decay length of the correlator diverges at the critical te... | 2 | https://mathoverflow.net/users/11260 | 376521 | 157,012 |
https://mathoverflow.net/questions/376492 | 8 | $\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $\mathcal{P}\_{n-1}$ be the space of complex polynomials in one variable, say $z$, of degree at most $n-1$. As a complex vector space, it is clearly $n$-dimensional. Consider the basis $1$, $z,\ldots,z^{n-1}$ of $\mathcal{P}\_{n-1}$. This allows us to identify $\mathcal... | https://mathoverflow.net/users/81645 | What is the subgroup of $\mathrm{SL}(n,\mathbb{C})$ which preserves the discriminant? | At the request of the OP, I post my comment as an answer. View $\mathcal{P}\_n$ as the space of homogeneous polynomials of degree $n$ in 2 variables. Let $\Delta \_p\subset \mathcal{P}\_n$ be the locus of polynomials with one linear factor of multiplicity $\geq p$. One can show that
the singular locus of $\Delta \_p$ i... | 9 | https://mathoverflow.net/users/40297 | 376528 | 157,014 |
https://mathoverflow.net/questions/376518 | 0 | Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a constant $C>0$ such that:
$$\int\_{\mathbb{R}^n}\frac{|u(x)-u(y)|}{|x-y|^{n+2s}}\,dy\leq C,\qquad\forall x\in\Omega,$$
wit... | https://mathoverflow.net/users/167027 | Uniform estimation of an integral | $\newcommand\ep\epsilon\newcommand\Om\Omega\newcommand\al\alpha\newcommand\R{\mathbb R}$According to your comment, $C^{0,2s+\ep}(\Om)$ is the set of all functions on $\Om$ that are Hölder-continuous with exponent $2s+\ep\in(0,1)$. It also appears that $\ep>0$ and that you extend the functions $u\in C^{0,2s+\ep}(\Omega)... | 2 | https://mathoverflow.net/users/36721 | 376532 | 157,015 |
https://mathoverflow.net/questions/376145 | 0 | Let $f:\mathbb{T}^m \to \mathbb{R}$ is a function of bounded variation(BV). Let $D=\{\boldsymbol{p}\_i,i=1,2,3\ldots\}$ be a countable dense subset of $(0,1)^m$. Let $E\_n, n = 1,2,3\ldots$ be a sequence of sets defined as $E\_n = \{\boldsymbol{p\_i}/\boldsymbol{p\_i}\in D, i = 1,2,3\ldots n\}$.
Define the mesh norm ... | https://mathoverflow.net/users/134538 | Estimate for computing the $L^2$-norm of a function from its data | Assume $m=1$ and $f$ is of bounded variation on $[0,1]$. The problem is to estimate
$$
\|f\|^2\_{2}-\frac{1}{n}\sum\limits\_{i=1}^n\left(f({p\_i})\right)^2=\int\_{0}^{1}f^{2}(t)dt
-\frac{1}{n}\sum\limits\_{i=1}^n f^{2}({p\_i}),
$$
as the number of points grows. Setting $g=f^{2}$, which is also of bounded variation, the... | 1 | https://mathoverflow.net/users/89429 | 376541 | 157,018 |
https://mathoverflow.net/questions/376531 | 2 | This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, *On topological cyclic homology*, arXiv:[1707.01799](https://mathoverflow.net/questions/376490/computing-homotopy-colimit-of-a-space-with-free-s1-action) (on last line p147), which I am having fundamental confusion.
---
We have the categories $\Lambda... | https://mathoverflow.net/users/97321 | Equivariant colimit and equivariant functors | Q1: For any $C,D\in Fun(BG,Cat\_\infty)$, $Fun(C,D)$ acquires a $G$-action too. Informally, this is described as $F\mapsto gF(g^{-1}-)$, and this is in fact an accurate description if $G$ is a discrete group and $C,D$ are $1$-categories; but more generally, formally you can see it as an internal hom in $Fun(BG,Cat\_\in... | 4 | https://mathoverflow.net/users/102343 | 376543 | 157,020 |
https://mathoverflow.net/questions/376270 | 0 | I quote [Delbaen and Shirakawa (2002)](https://link.springer.com/article/10.1023/A:1024125430287).
>
> Starting from a stochastic differential equation of the form:
> $$dr\_t=\alpha\left(r\_{\mu}-r\_t\right)dt+\beta\sqrt{\left(r\_t-r\_m\right)\left(r\_M-r\_t\right)}dW\_t\tag{1}$$
> with $\left\{W\_t\right\}\_{t\geq... | https://mathoverflow.net/users/165759 | Hitting probability for mean-reverting stochastic process | As the OP suggests, the confusion appears to be due to a typo in (3) and (4). Here are the corrected limits. \begin{align}
\tag{$\star$}
\rho\_{x,0} &= \lim\_{y \downarrow 0, \color{red}{z \uparrow 1}} \frac{B\_{x,z}(p,q)}{B\_{y,z}(p,q)} \;, \quad
\rho\_{x,1} = \lim\_{y \downarrow 0, \color{red}{z \uparrow 1}} \frac{B... | 1 | https://mathoverflow.net/users/64449 | 376544 | 157,021 |
https://mathoverflow.net/questions/376546 | 2 | Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a constant $C>0$ such that:
$$\int\_{\mathbb{R}^n}\frac{|u(x)-u(y)|}{|x-y|^{n+2s}}\,dy\leq... | https://mathoverflow.net/users/167027 | Uniform estimation of an integral involving a Hölder-continuous function | $\newcommand\ep\epsilon\newcommand\Om\Omega\newcommand\al\alpha\newcommand\R{\mathbb R}$Your desired conclusion is true. Indeed, take any $u\in C^{0,s}(\R^n)$ such that $u$ is Hölder-continuous on $\Om$ with exponent $2s+\ep\in(0,1)$. Then $u$ is continuous on $\R^n$ (which is all we need in place of the condition $u\i... | 3 | https://mathoverflow.net/users/36721 | 376547 | 157,023 |
https://mathoverflow.net/questions/376503 | 1 | Let $E$ be a $\mathbb R$-Banach space, $\mathcal M\_1(E)$ (resp. $\mathcal M\_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi\_\mu$ denote the characteristic function of $\mu\in\mathcal M\_1(E)$, $$\mathcal C\_1(E):=\left\{\varphi\_\mu:\mu\in\mathc... | https://mathoverflow.net/users/91890 | Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$ | A quick Google search on "infinitely divisible" and "Banach space" leads to Linde's *Probability in Banach Spaces: Stable and Infinitely Divisible Distributions* (John Wiley & Sons, 1986). There we find:
* Proposition 5.1.1: If $\mu$ is infinitely divisible on $E$, then $\hat\mu(a) \ne 0$ for every $a \in E'$.
* Coro... | 2 | https://mathoverflow.net/users/108637 | 376549 | 157,024 |
https://mathoverflow.net/questions/376548 | 4 | Let $A$ and $B$ be $C^\*$-algebras and let $A \otimes B$ their minimal tensor product and $M(A \otimes B)$ the associated multiplier algebra.
On $M(A \otimes B)$, we consider the strict topology which is the locally convex topology generated by the seminorms
$$M(A\otimes B)\ni x\mapsto \|x c\| $$
$$M(A \otimes B)\ni ... | https://mathoverflow.net/users/nan | Is the unit ball of $A \odot B$ strictly dense in that of $M(A \otimes B)$? | Yes. This follows from the strict topology version of the Kaplansky Density Theorem (the proof is much the same as the usual proof of Kaplansky Density). See for example Proposition 1.4 in Lance's book about Hilbert $C^\*$-modules. In multiplier algebra language it says:
**Theorem:** Let $A$ be a $C^\*$-algebra and l... | 3 | https://mathoverflow.net/users/406 | 376552 | 157,027 |
https://mathoverflow.net/questions/376530 | 13 | Let $G$ be a graph which does not contain a simple cycle $v\_1\ldots v\_k$ and two "crossing" chords $v\_iv\_j$ and $v\_pv\_q$, $i<p<j<q$. An example of such graph is a triangulation of the convex polygon. Is it true that the number of edges in $G$ does not exceed $2n-3$, where $n$ denotes the number of vertices?
| https://mathoverflow.net/users/4312 | Graph in which no cycle has two crossing chords | Thomassen and Toft [[JCTB 31(2):199-224, 1981](https://doi.org/10.1016/S0095-8956(81)80025-1)] showed that any graph with minimum degree at least 3 contains a cycle with two crossing chords from neighbouring vertices on the cycle. The $2n-3$ upper bound follows by induction on $n$, since we may delete a vertex of degre... | 12 | https://mathoverflow.net/users/25980 | 376557 | 157,030 |
https://mathoverflow.net/questions/376328 | 5 | For $k > 1$, is it possible that $\begin{pmatrix} a\_1 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} a\_2 & 1 \\ -1 & 0 \end{pmatrix}\ldots \begin{pmatrix} a\_k & 1 \\ -1 & 0 \end{pmatrix} = \pm \begin{pmatrix} b & 1 \\ -1 & 0 \end{pmatrix}$ if $a\_1,a\_2,\ldots a\_k,b$ are Eisenstein integers and $|a\_i| > 1$ for $i=1,2,... | https://mathoverflow.net/users/62575 | Can a product of Cohn matrices over the Eisenstein integers with non-zero, non-unit coefficients be a Cohn matrix? | To my surprise, not only is there a solution for *some* $b$, there is actually a very simple infinite family of solutions for *every* $b$. Let $\omega = \frac{1 + \sqrt{-3}}{2}$. Then
$\begin{pmatrix} a\_0 + a\_1\omega & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 2 -\omega & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 +... | 3 | https://mathoverflow.net/users/62575 | 376562 | 157,032 |
https://mathoverflow.net/questions/376561 | 5 | As stated, I wonder if there is a closed form for the generating function $F\_{\alpha,\beta}(x):=\sum\_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ where $\alpha,\beta \in\mathbb{N}$. Calling this a generating function is slightly misleading since ${\alpha \choose j}=0$ when $j>\alpha$ so this is really a fin... | https://mathoverflow.net/users/159298 | Closed form for $\sum_{j=0}^{\infty}{\alpha \choose j} {\beta \choose j}x^j$ | WolframAlpha immediately gives hypergeometric form $F\_{\alpha, \beta}(x) = {}\_2 F\_1(-\alpha, -\beta; 1; x)$.
| 7 | https://mathoverflow.net/users/106512 | 376564 | 157,033 |
https://mathoverflow.net/questions/376558 | 4 | I want to see if this series converges or not:
$$
\sum\_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2).
$$
I tried comparison tests but nothing. I saw that integral criteria works but I don't know how to show that.
Thank you
| https://mathoverflow.net/users/168752 | How to show this series converges $\sum\limits_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2)$ | As indicated by Todd Trimble in comments, we can use the Dirichlet test; here, since $$\sin(n)\sin(n^2)=\frac12\big( \cos n(n-1) - \cos n(n+1) \big)$$
we have a telescopic sum $$\sum\_{n=1}^M \sin(n)\sin(n^2)=\frac12-\frac12 \cos M(M+1)=\sin^2\Big(\frac{M(M+1)}2\Big),$$
that does not exceed $1$ in absolute value.
| 10 | https://mathoverflow.net/users/6101 | 376567 | 157,036 |
https://mathoverflow.net/questions/376596 | 1 | The classical finite field Kakeya conjecture state as following (for conveinent, all version of kakeya conjecture is state in hasdorff dimension version):
$\mathrm{Finite\ Field\ Kakeya\ Conjecture}$: Let $\mathrm{F}=\mathrm{F}/\mathrm{qF}$ be a finite field, let $K \subseteq \mathrm{F}^{\mathrm{n}}$ be a Kakeya se... | https://mathoverflow.net/users/114101 | Is finite field version kakeya conjecture still true when changing the line of every direction with only 2(or several but not the full line)element? | In his paper "[On the size of Kakeya sets in finite fields](https://arxiv.org/pdf/0803.2336.pdf)" (where the proof of the finite field Kakeya conjecture has appeared), Dvir also introduces the notion of a $(\delta,\gamma)$-Kakeya set, which is, essentially, a set $K\subset\mathbb F\_q^n$ with the following property: th... | 2 | https://mathoverflow.net/users/9924 | 376599 | 157,044 |
https://mathoverflow.net/questions/376606 | -1 | Any idea on whether or not $$\sum\_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$
has a closed formula on $a$ and $b$ (and on what it is, in case it does)?
It is supposed that $b \le a$.
| https://mathoverflow.net/users/131790 | Binomial Coefficients sum | Here's a combinatorial proof that the expression is $0$. Both sides of the identity count the number of $(b-1)$-subsets of $\{1,\dots,a+b-1\}$ that include $\{1,\dots,b\}$. Because $b > b-1$, this count is obviously $0$, establishing the RHS. For the LHS, apply inclusion-exclusion, where the $b$ properties to be avoide... | 2 | https://mathoverflow.net/users/141766 | 376614 | 157,048 |
https://mathoverflow.net/questions/376608 | 0 | Consider a function **fixed** function $f\in L^1(\mathbb{R})$ such that $$
\int\_{\mathbb{R}}f(x)dx=0
$$
Now define the following function: $$
F(y)=\int\_{\mathbb{R}} f(x)\mathrm{sech}\Big(\frac{x}{\exp(y)}\Big)dx.
$$
Then, by definition of $F$ is clear that $$
\lim\_{y\to\infty}F(y)=0.
$$
I am wondering if it is possi... | https://mathoverflow.net/users/159747 | Rate of convergence | $\newcommand\de\delta\newcommand\R{\mathbb R}$The answer is no: without any additional hypothesis on $f$, no such rate of convergence exists.
Indeed, suppose the contrary: that
$$|F(f)(y)|\le\de(y)\tag{1}$$
for some some function $\de\colon\R\to\R$ such that $\de(z)\to0$ as $|z|\to\infty$, all $f\in L^1(\R)$ with $\i... | 1 | https://mathoverflow.net/users/36721 | 376616 | 157,049 |
https://mathoverflow.net/questions/376315 | 1 | If, for all $k\in\mathbb{N}$, $(x\_{i}^{k})\_{i=1}^{\infty}\in X^{\mathbb{N}}$ is normalized and $M$-basic and if, in addition, for all $k\leq i\_{1}<i\_{2}<\ldots$ the diagonal sequence $(x\_{i\_{k} }^{k})\_{k=1}^{\infty}\in X^{\mathbb{N}}$ is $M$-basic, then $(x\_{i}^{k})\_{i=1,k\in\mathbb{N}}^{\infty}$ is said to be... | https://mathoverflow.net/users/165007 | Asymptotic models and passing to sub-arrays | The definition of the asymptotic model is not correct. The asymptotic model doesn't have to be spreading. You are allowed to pass to subsequences in each row but not in columns. With that the answer to your question is negative. Take any array that generates an asymptotic model which is not spreading. Then the subarray... | 1 | https://mathoverflow.net/users/3675 | 376617 | 157,050 |
https://mathoverflow.net/questions/376615 | 0 | let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell\_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le 1;\, p,q\in P\rbrace$ the set of line segments connecting pairs of points.
How are the numbers $\operatorname{card}(\... | https://mathoverflow.net/users/31310 | Distribution of line segment intersections in random pointsets | Let $p\_1,\dots,p\_n$ be iid random points uniformly distributed in a region $R$. For a given (straight) line segment $[a,b]$ connecting points $a$ and $b$, let
$$h\_{a,b}(p,q):=1([p,q]\cap[a,b]\ne\emptyset).$$
Then the cardinality of the random set of all line segments connecting pairs of the random points $p\_1,\dots... | 2 | https://mathoverflow.net/users/36721 | 376620 | 157,053 |
https://mathoverflow.net/questions/376631 | 1 | I [asked](https://mathoverflow.net/questions/376606/binomial-coefficients-sum?noredirect=1#comment955008_376606) how to calculate $$\sum\_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find the value of $$\... | https://mathoverflow.net/users/131790 | An ambitiouser binomial coefficients sum | WolframAlpha claims it is
$$\frac{(-1)^{k + 1} (k + 1) (-a - b + k + 1) \binom{b}{k + 1} \binom{a + b - k - 2}{a - k - 1}}{a b},$$
and you can absorb the $k+1$ and $b$, yielding
$$\frac{(-1)^{k+1} (-a - b + k + 1) \binom{b-1}{k} \binom{a + b - k - 2}{a - k - 1}}{a}.$$
A little simpler:
$$\frac{(-1)^k b \binom{b-1}{k} \... | 2 | https://mathoverflow.net/users/141766 | 376633 | 157,058 |
https://mathoverflow.net/questions/376579 | 8 | I have been trying to find a function $f : \mathbb R \to \mathbb R$ such that $\lim\_{x \to c} f(x)$ exists when $c$ is irrational and the limit doesn't exist when $c$ is rational.
I tried variations of the Dirichlet function and Thomae's function, but I couldn't get anywhere.
I also tried proving that such a functio... | https://mathoverflow.net/users/168773 | Is there a real valued function whose limit exists only on irrational numbers? | Arrange rationals in a sequence $q\_n$, and set
$$f(x) = \sum\_{n = 1}^\infty 2^{-n} \mathbb{1}\_{[q\_n,\infty)}(x),$$
where
$$\mathbb{1}\_{[q\_n,\infty)}(x) = \begin{cases} 1 & \text{if $x \geqslant q\_n$,} \\ 0 & \text{if $x < q\_n$.} \end{cases} $$
In other words,
$$f(x) = \sum\_{n : q\_n \leqslant x} 2^{-n} .$$
By ... | 10 | https://mathoverflow.net/users/108637 | 376640 | 157,060 |
https://mathoverflow.net/questions/376628 | 4 | In the paper [Woronowicz - $C^\*$-algebras generated by unbounded elements](https://www.fuw.edu.pl/%7Eslworono/PDF-y/GENER.pdf), I read that the $\*$-strong operator topology on $B(H)$ and the strict topology on $B(H)$ coincide. I believe this means the following:
Consider the multiplier algebra $M(B\_0(H)) =B(H)$. V... | https://mathoverflow.net/users/nan | Strict topology and $*$-strong toppology on $B(H)$ coincide | Yeah, I don't think this is true. The two topologies do agree on bounded sets, as you have already observed.
For a counterexample, let $H$ be a separable, infinite-dimensional Hilbert space and let $\mathcal{I}$ be the set of all finite-dimensional subspaces of $H$, ordered by inclusion. For $E \in \mathcal{I}$ let $... | 5 | https://mathoverflow.net/users/23141 | 376655 | 157,065 |
https://mathoverflow.net/questions/376653 | 4 | Let $a,b,c$ be poistive integers,and such $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$,fine the all $a,b,c$ such
$$a^2+3b^2c^2=7^c$$
I'm not sure that this question has been studied, but I've been trying for a long time$(a,b,c\le 100)$, and there's only one set of solutions：$(a,b,c)=(2,1,1)$,But I can't prove it. I may need you... | https://mathoverflow.net/users/38620 | How to solve this equation $a^2+3b^2c^2=7^c$ | We work in $\mathbb{Z}[\omega]$ where $\omega=\frac{1+i\sqrt{3}}2$. It is a factorial ring, and we factorize both sides as $(a+i\sqrt{3}bc)(a-i\sqrt{3}bc)=(2+i\sqrt{3})^n(2-i\sqrt{3})^n$. Since $2+i\sqrt{3},2-i\sqrt{3}$ are prime (and coprime), the guy $a+i\sqrt{3}bc$ can not be divisible by both (otherwise it is divis... | 7 | https://mathoverflow.net/users/4312 | 376670 | 157,069 |
https://mathoverflow.net/questions/376627 | 2 | It is known that the bilinear Strichartz estimate
$$\Vert e^{it\Delta}u\_0\cdot \overline{e^{it\Delta}v\_0}\Vert\_{L\_t^2L^2(\mathbb{R}^d)}\lesssim\_{\delta} \Vert u\_0\Vert\_{H^{-1/2+\delta}(\mathbb{R}^d)}\Vert v\_0\Vert\_{H^{(d-1)/{2}-\delta}(\mathbb{R}^d)},\quad\delta>0$$
for the Schrodinger equation in dimension $d... | https://mathoverflow.net/users/54552 | A quadratic estimate for the Schrödinger equation | In general spatial dimensions $d\geq 2$, (non-endpoint) Strichartz for Schrodinger holds with
$$ \| e^{it \Delta} u\_0 \|\_{L^{q}\_t L^p\_x} \lesssim \| u\_0\|\_{L^2}$$
where $p\in (2, \frac{2d}{d-2})$ and $q = \frac{4p}{d(p-2)}$. To get $L^4$ in time you want
$$ q = 4 \iff p = \frac{2d}{d-1} \tag{1}$$
So in $d... | 2 | https://mathoverflow.net/users/3948 | 376687 | 157,072 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.