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 |
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https://mathoverflow.net/questions/424922 | 1 | Consider the class of simple connected n/2-regular graphs, n even. Are the maximum clique problem and/or maximum independent set problem NP-complete on such graphs? Is there any known result which would imply NP-completeness or, otherwise, how can it be proved/disproved? Any helpful information comment would be highly ... | https://mathoverflow.net/users/484343 | Problem NP-completeness on a specific graph class | The answer is **yes**.
By taking complements, the maximum clique problem can be reduced to the maximum independent set problem (MIS) on graphs with degree $n/2-1$.
It is NP-hard to approximate MIS to within a factor of $7/6$, even for bounded degree graphs. (see [On the Hardness of Approximating Minimum Vertex Cove... | 1 | https://mathoverflow.net/users/125498 | 424940 | 172,565 |
https://mathoverflow.net/questions/424938 | 1 | In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia [*Dedekind psi function*](https://en.wikipedia.org/wiki/Dedekind_psi_function). On the other hand I add the reference that Wi... | https://mathoverflow.net/users/142929 | On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function | It is standard that all prime factors of Fermat number $F\_m$ are of the form $2^{m+2}k+1$, in particular they are all at least $2^m$. It is then also clear that $F\_m$ can only be divisible by at most $2^m/m$ such primes. Therefore
$$\frac{\psi(F\_m)}{F\_m}=\prod\_{p\mid m}\left(1+\frac{1}{p}\right)<\left(1+\frac{1}{2... | 9 | https://mathoverflow.net/users/30186 | 424944 | 172,566 |
https://mathoverflow.net/questions/424908 | 1 | In my research in a different field (representation theory), the following system of equations popped up:
$$
ax=by
$$
$$
xy+a+b-ax=p
$$
where $p\in\{0,1,2,3,4\}$ and $a,b,x,y$ are integers (I am also interested in the case where x and y are rationals). I have found some solutions in some special cases e.g. when a=0... | https://mathoverflow.net/users/484326 | Solutions to a system of Diophantine equations | Here is solution in positive integers.
The equation $ax=by$ implies that there are four integers $u,v,w,t$ such that $a=uv$, $x=wt$, $b=uw$, $y=vt$.
Then the second equation takes form
$$vwt^2 + uv + uw - uvwt = p.$$
There are two cases:
**Case $u\leq t$.** We have
$$uv + uw \leq p,$$
which has a finite number of s... | 1 | https://mathoverflow.net/users/7076 | 424946 | 172,568 |
https://mathoverflow.net/questions/424941 | 1 | Finding UMVUE (Minimum-variance unbiased estimator) looks like an optimisation problem (minimise the variance of $\hat{\theta}$ given the constraint $\text{E}\_\theta\hat{\theta}=\theta.$
I tried to apply here a standard method from optimisation but failed.
The setting: $(X\_1,\dots, X\_n)$ are i.i.d. random variable... | https://mathoverflow.net/users/131858 | UMVUE as an optimization problem | $\newcommand\th\theta$
1. It is not true in general that the Cramér--Rao lower bound is a solution to a meaningful optimisation problem: ["Under some regularity conditions, the Cramér-Rao lower bound is
attained iff $f\_\th$ is in an exponential family" (p.14)](https://pages.stat.wisc.edu/%7Eshao/stat709/stat709-17.p... | 3 | https://mathoverflow.net/users/36721 | 424947 | 172,569 |
https://mathoverflow.net/questions/424929 | 2 | My question is the following:
* Given $x,y \in \omega^\omega$ such that $x\equiv\_c y$ is there an $L$-definable continuous map $\varphi: \omega^\omega\rightarrow \omega^\omega$ such that $\varphi(x) = y$?
By $x\equiv\_c y$ I mean that they are in the same constructibility degree, i.e. $L(x) = L(y)$ and by $\varphi... | https://mathoverflow.net/users/141146 | A continuous map relating co-constructible reals | *By "formula" I will mean "formula of set theory with ordinal parameters." Note that "ordinal parameters" is equivalent to "constructible parameters" for our purposes, since $L$ carries a definable bijection with $\mathsf{Ord}$.*
*Also, since the answer to your question is trivially true if $V=L$, I'm interpreting yo... | 3 | https://mathoverflow.net/users/8133 | 424954 | 172,572 |
https://mathoverflow.net/questions/424974 | 7 | Let $f\in S\_k(\Gamma\_0(N))$ be a cusp form for $N>1$. Consider the following operators acting on $f$ via the natural action of $GL\_2^{+}(\mathbb{R})$ :
$$ W\_N=\begin{pmatrix}
0 & -1\\
N & 0
\end{pmatrix}$$
$$ U\_q=\sum\limits\_{i=0}^{q-1}\begin{pmatrix}
q & i\\
0 & q
\end{pmatrix}$$ for prime $q\mid N$.
Does ... | https://mathoverflow.net/users/155716 | Fricke involution and Atkin operator | **EDIT.** In the answer below, $U\_q$ refers to the usual Hecke operator given on Fourier expansions by $\sum\_{n \geq 1} a\_n x^n \mapsto \sum\_{n \geq 1} a\_{qn} x^n$. The operator $U\_q$ in the OP is given by $\sum\_{n \geq 1} a\_n x^n \mapsto q \sum\_{n \geq 1} a\_{qn} x^{qn}$. As explained in the comments this doe... | 8 | https://mathoverflow.net/users/6506 | 424977 | 172,578 |
https://mathoverflow.net/questions/424978 | 0 | Assume that $\Phi, \Psi$ are positive increasing functions and $g$ positive non-increasing so that
$$\int\_0^1 \Phi\left(\frac{g(t)}{t}\right)dt = \int\_0^1 \Phi\left(\frac{1}{t}\right)dt=1.$$
Then it seems to me that $$\int\_0^1 \Phi\left(\frac{g(t)}{t}\right)\Psi(t)dt\le \int\_0^1 \Phi\left(\frac{1}{t}\right)\Psi(t... | https://mathoverflow.net/users/409893 | An integral inequality revisited | Choose $a\in [0,1]$ such that $g(t)\geqslant 1$ for $t\leqslant a$ and $g(t)\leqslant 1$ for $t\geqslant a$. Then $(\Phi(g(t)/t)-\Phi(1/t))(\Psi(t)—\Psi(a))\leqslant 0$ for all $t\in [0,1] $. Integrate it.
| 2 | https://mathoverflow.net/users/4312 | 424981 | 172,580 |
https://mathoverflow.net/questions/424980 | 5 | Let $(M,g)$ be a Riemannian manifold, endowed with the Levi-Civita connexion $\nabla$ induced by $g$. By the very definition of the Levi-Civita connexion $\nabla$, we indeed know that $\nabla g=0$, i.e., the (total) covariant derivative of the metric tensor vanishes. Now assume that $G$ is another metric tensor field o... | https://mathoverflow.net/users/484386 | The vanishing of covariant derivative of an alternative metric tensor | In an irreducible Riemannian manifold $(M,g)$, every symmetric tensor that satisfies $\nabla^{g}T=0$ must be of the form $T=kg$ for some constant $k$. See Theorem 10.3.2 in Chapter 10 of Peter Peterson's *Riemannian geometry, 3rd Edition*. It is shown there how to use this fact to conclude that irreducible symmetric sp... | 10 | https://mathoverflow.net/users/144247 | 424984 | 172,581 |
https://mathoverflow.net/questions/424983 | 7 | Let G be an infinite group wich is finitely generated.
Is that true that the size of all finite conjugacy classes is bounded?
What I know. If G is a finitely generated FC-group then it's true (follows from [this](https://groupprops.subwiki.org/wiki/Finitely_generated_and_FC_implies_FZ)). But if G isn't FC- or FZ-gr... | https://mathoverflow.net/users/156446 | Size of conjugacy classes in infinite groups | The answer is no: there exists a 2-generated group, having finite conjugacy classes of unbounded size.
Indeed B.H. Neumann (1937) produced a 2-generated group $G$ with normal subgroups $(H\_n)\_{n\ge 5}$ such that $H\_n\simeq \mathrm{Alt}\_n$. Since $H\_n$ has a conjugacy class of size growing to infinity (say, the s... | 8 | https://mathoverflow.net/users/14094 | 424986 | 172,583 |
https://mathoverflow.net/questions/424970 | 8 | Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. Let $\mathcal L$ be the set of linear functions $\mathbb R \to \mathbb R$.
Define the roughness $\mathcal Rf(x)$ of $f$ at $x \in \mathbb R^n$ by
$$\inf\_{L \in \mathcal L} \limsup\_{y \to x} \left | \frac{f(y) - f(x) - L(y-x)}{|y - x|} \right |.$$
We s... | https://mathoverflow.net/users/173490 | Pseudo differentiable functions | The function
$$
f(x) = \begin{cases} x\sin 1/x^2 & x\not= 0 \\ 0 & x=0 \end{cases}
$$
gives a counterexample. We have $f\in C^{\infty}(U)$ when we restrict to $U=\mathbb R\setminus \{ 0\}$, so if $f$ had a distributional derivative in $L^1\_{\textrm{loc}}$, it would have to be its classical derivative $f'=-(2/x^2)\cos ... | 8 | https://mathoverflow.net/users/48839 | 424993 | 172,588 |
https://mathoverflow.net/questions/424972 | 1 | Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{a} \subset R$ be an ideal. Let
$$ \mathfrak{b} = \bigcap \{R \cap \mathfrak{a} \cdot R\_\mathfrak{p} \text{ } \colon \mathfrak{p} \in \text{Ass}(R/\mathfrak{a}) \text{ and } \mathfrak{m} \neq \mathfrak{p} \}. $$
Is it true that $R/\mathfrak{b}$ is a C... | https://mathoverflow.net/users/113200 | Cohen-Macaulay quotient ring and symbolic power | I don't think so. Take $R=k[[x,y,z,w]]$ and take $\mathfrak{a}=(x,y)\cap (z,w)$. Then $\operatorname{Ass}(R/\mathfrak{a})=\{(x,y), (z,w)\}$, and $R\cap \mathfrak{a}\cdot R\_P=P$, so $\mathfrak{b}=\mathfrak{a}$, but $R/\mathfrak{a}$ is not Cohen-Macaulay since its vanishing locus is two planes that meet at a point. Do y... | 1 | https://mathoverflow.net/users/66536 | 424996 | 172,591 |
https://mathoverflow.net/questions/425013 | 5 | Let $Q : \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial. Form the set
$$M\_{Q} := \{p:\text{ }p\text{ is prime, }\exists n\_{p}\in \mathbb{Z}\text{ so that }p|Q(n\_{p})\}$$
Is $$\sum\_{s \in M\_{Q}}\frac{1}{s} = \infty ?$$
This question is asked so that one can somewhat understand the density of primes dividi... | https://mathoverflow.net/users/134295 | Divergence of primes dividing polynomials | Yes, the series diverges. We can reduce easily to the case of irreducible monic $Q$.
Next, let $\alpha\_Q(p)$ be the number of roots of $Q(x)$ in $\mathbb{Z}/p\mathbb{Z}$. Note that $M\_Q$ is the set of primes $p$ for which $\alpha\_Q(p)$ is positive. Landau's Prime Ideal Theorem (applied to the field $K=\mathbb{Q}(\... | 10 | https://mathoverflow.net/users/31469 | 425015 | 172,598 |
https://mathoverflow.net/questions/424955 | 3 | Recall that a *numerical semigroup* $S$ is a submonoid of the non-negative integers $\mathbb Z\_{\geq 0}$ whose relative complement $\mathbb Z\_{\geq 0} \setminus S$ is finite. Observe that the collection $S^\*$ of nonzero elements of $S$ constitutes an *ideal* of $S$ in the sense that $$S^\* \supseteq \{s + t \mid s \... | https://mathoverflow.net/users/160770 | On the Hilbert function of a numerical semigroup | I believe there is an easier proof.
**Proof.** By induction, it suffices to prove the case that $eS^\* = e + (e - 1) S^\*$: indeed, if we assume that $(n + 1) S^\* = e + nS^\*$ for some integer $n \geq e,$ then it holds that $$(n + 2) S^\* = (n + 1) S^\* + S^\* = e + nS^\* + S^\* = e + (n + 1) S\*.$$
Because the co... | 2 | https://mathoverflow.net/users/160770 | 425021 | 172,601 |
https://mathoverflow.net/questions/424863 | 6 | In his paper "Symmetric products of the circle" (1966) H. R. Morton proves among other things that the usual multiplication
\begin{align\*}
\text{SP}^n(S^1)&\to S^1 \\
[x\_1,\dots,x\_n]&\mapsto x\_1\cdots x\_n
\end{align\*}
is a fibration with fiber $\Delta^{n-1}$. Such a map can still be definied if one replaces $S^1$... | https://mathoverflow.net/users/474147 | Fibration from symmetric product to initial space | Let $A$ be a topological abelian group. One condition that guarantees that the map $\mu\_n\colon\operatorname{SP}^n(A)\to A$ is a fiber bundle is that $A$ is "locally divisible by $n$". More precisely, suppose there exists an open neighborhood $U$ of the identity, together with a continuous function $U\to A$ that behav... | 5 | https://mathoverflow.net/users/6668 | 425024 | 172,603 |
https://mathoverflow.net/questions/425002 | 3 | It is well known that [the $p$-norms tend to the $\infty$-norm](https://math.stackexchange.com/q/242779/791850), in that if $\lVert f \rVert\_q < \infty$ for some $q \ge 1$ then $\lVert f \rVert\_p \to \lVert f \rVert\_\infty$ as $p \to \infty$. Does this extend in some way to [general Orlicz norms](https://en.wikipedi... | https://mathoverflow.net/users/158681 | When do Orlicz norms tend to the uniform norm? | $\newcommand{\ep}{\varepsilon}\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}$A natural generalization of the fact that the $p$-norm converges to the $\infty$-norm as $p\to\infty$ is as follows.
Let $\mu$ be a probability measure on a measurable space $(S,\Si)$. For each natural $n$, let $\Phi\_n$ be a [Young func... | 4 | https://mathoverflow.net/users/36721 | 425039 | 172,606 |
https://mathoverflow.net/questions/424931 | 2 | I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The question concerns the relationship between various definitions of overconvergent modular forms in the standard papers... | https://mathoverflow.net/users/165625 | Overconvergent modular forms and the level at $p$ | The curve $X\_1(Np^n)$ is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union of several components. What you want for a good theory of overconvergent forms is to pick out just one of these components. This ... | 2 | https://mathoverflow.net/users/2481 | 425053 | 172,608 |
https://mathoverflow.net/questions/425057 | 8 | I have been considering the following question:
Let $X$ be a compact, metrizable space with the following property: every (regular) Borel probability measure on $X$ is atomic, i.e. for each $\mu\in\text{Prob}(X)$ there exists $x\_\mu\in X$ such that $\mu(\{x\_\mu\})>0$. Does it follow that $X$ is countable?
It is t... | https://mathoverflow.net/users/484440 | The class of spaces where every Borel measure is atomic | Yes. Every uncountable Polish space is isomorphic as a measurable space to the unit interval by Kuratowski's isomorphism theorem and admits, therefore, a nonatomic probability measure. On a Polish space, every finite Borel measure is regular.
| 9 | https://mathoverflow.net/users/35357 | 425059 | 172,610 |
https://mathoverflow.net/questions/425051 | 1 | Let $L:\mathcal{M}\leftrightarrows\mathcal{N}:R$ be a Quillen equivalence between combinatorial model categories such that all objects are fibrant. Let $X$ be a cofibrant object of $\mathcal{M}$. Then the unit of the adjunction gives rise to a weak equivalence $X\to RL(X)$.
>
> Is there a known sufficient condition... | https://mathoverflow.net/users/24563 | Unit of a Quillen equivalence and fibration | If we write down the lifting square for an arbitrary cofibration $f\colon A→B$ and the unit map $η\colon X→RLX$ (with the bottom map being $b\colon B→RLX$),
and then use the adjunction to pass to the adjoint square with maps $Lf$ and $\def\id{{\rm id}} \id\_{LX}$, the resulting lifting problem has a solution if and onl... | 1 | https://mathoverflow.net/users/402 | 425062 | 172,611 |
https://mathoverflow.net/questions/425047 | 11 | Wilson's theorem (actually proven by Lagrange) from elementary number theory states that: If $n\ge 2$ is an integer, then
$$
(n-1)! \equiv
\begin{cases}
\hfill -1 \pmod {n} &\text{ if } n \text{ is prime}\\
\hfill 2 \pmod {n} &\text{ if } n=4\\
\hfill 0 \pmod {n} &\text{ if } n \text{ is composite, } n\ne 4
\end{cas... | https://mathoverflow.net/users/31084 | Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist? | I will show the two results are non-superficially related by showing one of them implies the other: the classification of moduli $n \geq 2$ for which the unit group $(\mathbf Z/(n))^\times$ is cyclic implies Gauss' generalization of Wilson's theorem.
The proof is presented in three steps. All the basic ideas are pres... | 11 | https://mathoverflow.net/users/3272 | 425076 | 172,614 |
https://mathoverflow.net/questions/424958 | 6 | Let $0<a\_1 \le a\_2 \le \cdots \le a\_n$ be positive integers such that $a\_1 + \cdots + a\_n = m$ and $\gcd(a\_1,\ldots,a\_n)=1$. Let $\mathbf a :=(a\_1,\ldots,a\_n)\in\mathbb Z^n$ and $\mathbf x:=(x\_1,\ldots,x\_n)\in\mathbb Z^n$. Consider the equation
$$
\mathbf a\cdot \mathbf x = a\_1 x\_1 + \cdots + a\_n x\_n \eq... | https://mathoverflow.net/users/149337 | Bounds on Bézout coefficients | We may suppose that $-m<k\leqslant 0$ and choose integers $y\_i$ such that $\sum y\_ia\_i=k$. Next, by replacing $(y\_1,y\_i)\to (y\_1\pm a\_i, y\_i\mp a\_1)$ we may achieve $y\_i\in [0,a\_1)$ for all $i>1$. Then $y\_1=(k-\sum\_{i>1}y\_ia\_i)/a\_1\in (-2m,0]$. Denote $t=\lceil n/2\rceil$. We have $a\_i\leqslant 2m/n$ f... | 2 | https://mathoverflow.net/users/4312 | 425083 | 172,616 |
https://mathoverflow.net/questions/425061 | 7 | I was inspired by [Does there exist a function which converts exponentiation into addition?](https://math.stackexchange.com/questions/4475785/does-there-exist-a-function-which-converts-exponentiation-into-addition) to think about mapping exponentiation onto addition.
The question asks whether there exists $f:\mathbb{R}... | https://mathoverflow.net/users/171026 | Mapping exponentiation onto addition | $\def\abs#1{\lvert#1\rvert}$Though it’s not clearly stated in the question, I take it $f$ is supposed to be defined only for $a>0$, as otherwise $a^b$ has no sensible definition for non-integer $b$.
For $a>0$, $a\ne1$ and $b\ne0$, a simple solution is
$$\log\abs{\log a^b}=\log\abs{\log a}+\log\abs b.$$
The domain r... | 13 | https://mathoverflow.net/users/12705 | 425088 | 172,618 |
https://mathoverflow.net/questions/425090 | 2 | I would like to know of any examples of families of groups that are known (or conjectured) to have a solvable uniform word problem, i.e. an algorithm that given a presentation $P$ of a group in the family, and a word $W$ in the generators of $P$, decides whether $W$ represents the identity of the group defined by $P$.
... | https://mathoverflow.net/users/69681 | Examples of group families with solvable uniform word problem | Derek Holt is the expert here - I hope he will correct me where I err.
---
There are many such families. Here are a few ones that spring to mind (or were mentioned in the comments above), in no particular order.
1. fuchsian groups
2. fundamental groups of three-manifolds
3. hyperbolic groups
4. automatic groups... | 3 | https://mathoverflow.net/users/1650 | 425098 | 172,620 |
https://mathoverflow.net/questions/424906 | 0 |
>
> B is a b-open set if $B\subset Cl(IntB) \cup Int(ClB)$
>
>
>
>
> A topological space $X$ is b-disconnected if it can be expressed as a union of two disjoint non-empty b-open sets. Otherwise, $X$ is said to be b-connected.
>
>
>
Earlier I asked on MSE about b-connected, b-disconnected as well as totall... | https://mathoverflow.net/users/59205 | Examples of b-connected sets? | Notice that if a set $S$ is a regular-closed subset of the space $X$ (that is, if $S$ is the closure of an open subset of $X$), then $S$ is b-open. Furthermore, any open subset is b-open. Therefore, if $X$ has a non-empty proper regular-closed set $S$, then $S$ and $X \setminus S$ shows that $X$ is b-disconnected. In p... | 2 | https://mathoverflow.net/users/89233 | 425100 | 172,621 |
https://mathoverflow.net/questions/425081 | 5 | Given a linear order $\mathcal{S}$, let $\mathbb{A}\_\mathcal{S}$ be the class of all ordertypes which do not embed $\mathcal{S}$ (= do not have a suborder isomorphic to $\mathcal{S}$). Say that a linear order $\mathcal{S}$ is **deep** iff there is a computable functional $\Phi$ with the property that, whenever $\mathc... | https://mathoverflow.net/users/8133 | Computable functionals avoiding embeddings of linear orderings | Yes, $\zeta$ is deep. Observe that $\mathcal{M} \in \mathbb{A}\_\zeta$ iff $\mathcal{M} = L\_0 + L\_1$ for some $L\_0$ well-founded and $L\_1$ reverse well-founded. Moreover, this division is arithmetical in $\mathcal{M}$: to determine which side a point $x$ belongs to, search for $x+1$, $x-1$, $x+2$, $x-2$, etc (where... | 2 | https://mathoverflow.net/users/32178 | 425101 | 172,622 |
https://mathoverflow.net/questions/424982 | 4 | Let $X$ be the Grassmannian variety $\operatorname{Gr}(k,n)$ of $k$-planes in $\mathbb{C}^n$. I'm aware of two ways to describe its $T$-equivariant cohomology:
1. (Quotient ring) $H\_T^\*(X)=\Lambda[e\_1(x|t),\dotsc,e\_k(x|t)]/(h\_{n-k+1}(x|\dotsc,h\_{n}(x|t))$
where $\Lambda=\mathbb{C}[t\_1,\dotsc,t\_n]$, and $e\_i(... | https://mathoverflow.net/users/138150 | Explicit isomorphism between two realizations of $H^*_T(\operatorname{Gr}(k,n))$ (reference request) | $\def\Fl{\mathcal{F}\ell}$I'd been hoping someone else would answer this, since I don't have a good reference to cite and I tend to make minor errors in things like this, but some answer is better than none. As in your question, $\Lambda := H\_T^{\ast}(\text{point}) \cong \mathbb{Z}[t\_1, t\_2, \ldots, t\_n]$.
Let's ... | 4 | https://mathoverflow.net/users/297 | 425104 | 172,623 |
https://mathoverflow.net/questions/425102 | 1 | I am currently working on my undergraduate thesis, and my adviser suggested that I look into a Polyhedral Active Set Algorithm (PASA) for my paper. I have been trying to find resources/materials on it online, but most of the papers I have seen and read unfortunately seem to be accessible only to graduate students. So, ... | https://mathoverflow.net/users/476977 | Resources/Reading Materials on PASA (optimal control theory) | The 2022 article [A Gradient-Based Implementation of the Polyhedral Active Set Algorithm](https://arxiv.org/abs/2202.05353) discusses one particular PASA implementation in much detail --- it does not seem to require much by way of background knowledge (other than familiarity with conjugate gradient algorithms).
You c... | 1 | https://mathoverflow.net/users/11260 | 425107 | 172,624 |
https://mathoverflow.net/questions/425106 | 2 | Related to the [question about a(n)=a(n-1)+a(floor(n/2))](https://mathoverflow.net/questions/401272/bounds-for-an-an-1a-lfloor-n-2-rfloor)
Let $A$ be real constant $ 0 < A < 1$.
Define the sequence $a(n)$ by $a(1)=1, a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$
(if you prefer take $a'(n)=a'(n-1)+a'(n-\lfloor n^A \rfloor... | https://mathoverflow.net/users/12481 | Bounds for the sequence $a(n)=a(n-1)+a(\lfloor n-n^A \rfloor)$ | Let us show that
\begin{equation\*}
a(n)\le\exp(n^{1-A+o(1)}) \tag{1}\label{1}
\end{equation\*}
(as $n\to\infty$).
Indeed, for each $q\in(1-A,1)$,
\begin{equation\*}
(k-1)^q-k^q\sim-qk^{q-1},\quad (k-k^A)^q-k^q\sim-qk^{A+q-1}
\end{equation\*}
(as $k\to\infty)$, so that
\begin{equation\*}
\frac{\exp((k-1)^q)+\exp((... | 4 | https://mathoverflow.net/users/36721 | 425113 | 172,626 |
https://mathoverflow.net/questions/425114 | 0 | Let $X$ be a smooth projective variety of dimension $n$ and $i: D\hookrightarrow X$ be a smooth, anti-canonical divisor in $X$. In other words,$[D]+[\omega\_X]=[0]$ where $\omega\_X$ is the canonical bundle of $X$.
Let $i^\*: D^b\_{coh}(X)\to D^b\_{coh}(D)$ and $i\_\*: D^b\_{coh}(D)\to D^b\_{coh}(X)$ be the derived p... | https://mathoverflow.net/users/24965 | What is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i: D\hookrightarrow X$? | For any Cartier divisor $i \colon D \hookrightarrow X$ and any $F \in D^b(X)$ there is an exact triangle
$$
F \otimes \mathcal{O}\_X(-D) \to F \to i\_\*i^\*F.
$$
It can be obtained by tensoring the exact sequence
$$
0 \to \mathcal{O}\_X(-D) \to \mathcal{O}\_X \to i\_\*\mathcal{O}\_D \to 0
$$
with $F$ and using the proj... | 2 | https://mathoverflow.net/users/4428 | 425115 | 172,627 |
https://mathoverflow.net/questions/425119 | 4 | We know that given a sufficiently regular function $f: \mathbb{R} \to \mathbb{R}$, then its periodisation (say to period $1$) is given by
$$
\begin{align}
F(x) := \sum\_{n\in\mathbb{Z}} f(x + n).\tag{$A$}
\end{align}
$$
Say instead we have a periodic function $F:[0,1] \to \mathbb{R}$ with period 1. Is it possible/is ... | https://mathoverflow.net/users/160454 | How to unperiodise a function | Take a periodic partition of unity, that is to say a test function $\chi(x)$ such that $\sum \chi(x+n)=1$, and define $f(x)=F(x)\chi(x)$.
In case you are wondering, a periodic partition of unity is easy to build: pick a non negative test function $\psi(x)$ which is strictly positive on $[0,1]$ and zero outside $[-1/2... | 6 | https://mathoverflow.net/users/7294 | 425120 | 172,628 |
https://mathoverflow.net/questions/425110 | 10 | Let $X$ be a scheme. Assume we have two closed subschemes $Y\_1$, $Y\_2$ that cover $X$ set-theoretically. Are there closed subschemes $Y'\_1$, $Y'\_2$ with the same underlying sets, such that the natural map $Y'\_1\amalg Y'\_2\to X$ is schematically dominant? (Equivalently, we want the intersection of the ideal sheave... | https://mathoverflow.net/users/7666 | Finite coverings by closed subschemes | Here's a counterexample.
$R=k[x,y\_1,y\_2,...]/((xy\_i)^{1+i}: i\ge1)$.
$I\_1=(x), I\_2=(y\_1,y\_2,...)$.
$I\_1,I\_2,I\_1 \cap I\_2$ are radical ideals and each element of $I\_1 \cap I\_2$ is nilpotent.
Some power of $x$, say $x^n$, is in $I'\_1$. Some power of $y\_n$, say $y\_n^m$, is in $I'\_2$. Then $x^ny\_n... | 10 | https://mathoverflow.net/users/59248 | 425121 | 172,629 |
https://mathoverflow.net/questions/425116 | 3 | I was wandering whether this weak form of $\text{CH}$ holds in $L(\mathbb{R})$ provably in $\text{ZF}+\text{DC}$
>
> $(\text{ZF}+\text{DC}) \ L(\mathbb{R})\vDash \forall X\subseteq\mathbb{R} ( X \text{ countable} \lor \mathcal{P}(X)\not\le^\* \mathbb{R})$
>
>
>
where $\mathcal{P}$ is the powerset and $x\le^\* ... | https://mathoverflow.net/users/141146 | Weak form of $\text{CH}$ in $L(\mathbb{R})$ | No, this can fail. Force $\mathrm{MA}\_{\omega\_1}$ over $L$ by ccc forcing and let $M$ be the $L(\mathbb R)$ of the extension $L[G]$. Note that $M$ has the same cardinals as $L$. We have $X=\mathbb R\cap L\in M$ is of size $\omega\_1$, however $M$ has a surjection $f:\mathbb R\rightarrow \mathcal P(\omega\_1)\cap M$: ... | 6 | https://mathoverflow.net/users/125703 | 425135 | 172,636 |
https://mathoverflow.net/questions/425134 | 0 | Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E\_{A}(\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1} A^T A))?$$
Any papers/resources would be helpful, ideal fidnings would be $m,n \mapsto \in... | https://mathoverflow.net/users/483386 | Approximating the expectation of trace inverse of random Gaussian combination | The answer is similar to that of your [earlier question](https://mathoverflow.net/q/424771/11260):
For $m,n\gg 1$, and $m/n\equiv r\in (0,1)$ fixed, an integration over the [Marchenko–Pastur distribution](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution) gives (with $x\_\pm=(1\pm\sqrt{r})^2$)
$$\li... | 1 | https://mathoverflow.net/users/11260 | 425137 | 172,638 |
https://mathoverflow.net/questions/425124 | 6 | H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, *Spin Geometry*, (1989), p. xi:
>
> ...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic operator canonically associated to the underlying riemannian metric. ...Twisting the Dirac-type operator...led...t... | https://mathoverflow.net/users/472967 | What were the "questions unapproachable by other means" w.r.t. $KO$-invariants? | Taken in the context of the introduction, I would guess that at least in part they are referring to applications of the index theorem to questions about positive scalar curvature. Before Lichnerowicz's application (1963) of the index theorem for the spin Dirac operator, it might have been conceivable that every manifol... | 6 | https://mathoverflow.net/users/3460 | 425142 | 172,639 |
https://mathoverflow.net/questions/425144 | 5 | I've seen the following sentence come up in a few papers:
>
> Consider the modular curve $Y\_1(N)$ and let $E$ be the universal elliptic curve over $Y\_1(N)$.
>
>
>
This comes up in Deligne's construction of Galois representations for modular forms of weight $k>2$. I'm not sure what "universal elliptic curve" ... | https://mathoverflow.net/users/394740 | Reference for universal elliptic curves | For any $n\geq 1$, one can define a functor $\mathcal{M}\_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a chosen point of exact order $n$; the morphisms in the groupoid are the isomorphisms of elliptic curves that respect the ch... | 4 | https://mathoverflow.net/users/2039 | 425155 | 172,643 |
https://mathoverflow.net/questions/425068 | 3 | Let's say I have a two odd primes, $p, q$ and $K$ is the field $\mathbb{Q}(\zeta\_{pq})$. Let's say $\alpha \in \mathcal{O}$ is an **arbitrary** element in the ring of integers of $K$, $\frak{b} \subset \mathcal{O}$ is a prime ideal of $\mathcal{O}$, and $\alpha \notin \frak{b}$. Not sure what it's called but I'd like ... | https://mathoverflow.net/users/138669 | Computing mth power residue symbols | I am not sure what you mean by "to resort to these symbols". If you want to compute their values given $\alpha$ and ${\mathfrak b}$, just use the definition.
Explicit reciprocity laws for higher powers do exist, but have a couple of natural drawbacks. If you look at the computation of $(\frac ab)$ in integers, you wi... | 5 | https://mathoverflow.net/users/3503 | 425161 | 172,646 |
https://mathoverflow.net/questions/297111 | 11 | Prompted by [this MO question](https://mathoverflow.net/questions/297049/is-eta-tau2-a-modular-form-of-weight-1-on-gamma12), I have the following question about modular forms which do not vanish on the upper-half plane.
**Q1.** Let $N \geq 1$ be an integer and let $\Gamma(N)$ be the principal congruence subgroup of $... | https://mathoverflow.net/users/6506 | Non-vanishing modular forms | I will answer Q2:
N=2: Denote by $Y^1(2)$ the moduli of elliptic curves with point of order 2 and fixed invariant differential. It is not hard to show that $Y^1(2) = \mathrm{Spec}\, \mathbb{Z}[\frac12][x\_2, y\_2, \Delta^{-1}]$ with $\Delta = 16x\_2^2y\_2^2(x\_2-y\_2)^2$. We obtain $Y(2)$ as the $\mathbb{G}\_m$-stack... | 2 | https://mathoverflow.net/users/2039 | 425163 | 172,648 |
https://mathoverflow.net/questions/425168 | 7 | Let $f:X\rightarrow Y$ be a proper embedding between complex manifolds, then the normal bundle $N$ is complex which is in paticular $\textsf{spin}^c$. Hence we have a Thom class $\lambda\_N$ and a Thom isomorphism
$$K^{\bullet}(X)\rightarrow \tilde{K}^{\bullet}(\text{Th}(N))$$
in topological $K$-theory. When $X,Y$ are ... | https://mathoverflow.net/users/nan | Does a Gysin map depend on the choice of Thom class? | Yes it does! The map in your display is (more or less) the cup product with the Thom class $\lambda\_N$. So if you choose a different Thom class you'll get a different map.
For example, take the standard inclusion $\mathbb{CP}^n\hookrightarrow\mathbb{CP}^{n+1}$. The normal bundle is the tautological bundle $L\rightar... | 9 | https://mathoverflow.net/users/163893 | 425192 | 172,657 |
https://mathoverflow.net/questions/425183 | 1 | Let $^\omega\omega$ be the collection of all functions $f:\omega\to\omega$. We order $^\omega\omega$ lexicographically, that is: For $f\neq g \in \,^\omega\omega$ let $m(f,g):= \min\{n\in\omega:f(n)\neq g(n)\}$, and then we say $f < g$ if and only if $f(m(f,g))<g(m(f,g))$.
This establishes a linear order $\leq\_{\tex... | https://mathoverflow.net/users/8628 | Embedding $^\omega\omega$ and $S_\omega$ with lexicographic order into $\mathbb{R}$ | There's a general argument we can use for any question of this form.
Cantor's theorem shows that every countable linear order embeds into $\mathbb{Q}$. Consequently, every *separable* linear order (= has a countable dense suborder) embeds into $\mathbb{R}$. This gives a positive answer to both your questions; for exa... | 5 | https://mathoverflow.net/users/8133 | 425194 | 172,658 |
https://mathoverflow.net/questions/425185 | 9 | $E\_6$'s Dynkin diagram is a line of 5 vertices, which we will number 1...5, and a sixth one attached to #3, which we will ignore.
$\dim V\_{\omega\_2} = 351 = \dim V\_{2\ \omega\_1}$, where $\omega\_i$ denotes the corresponding fundamental weight and $V$ the irrep with that high weight. (So $\dim V\_{\omega\_1} = 27... | https://mathoverflow.net/users/391 | Why do these two irreps of $E_6$ have the same dimension? | $\newcommand\Sym{\mathrm{Sym}}$
An extended comment which more or less suggests that your suggested answer might be as good as one can do.
If $G$ has a representation on $V$ which preserves a symmetric trilinear form on $V$, then $\Sym^3(V)$ has a $G$-invariant vector and thus $\Sym^2(V) \otimes V$ has a $G$-invari... | 6 | https://mathoverflow.net/users/484566 | 425195 | 172,659 |
https://mathoverflow.net/questions/424192 | 2 | I have questions about the proof of Theorem $2.1$ [here](https://arxiv.org/pdf/2105.11414.pdf). The proof is on Pg. $10$. I am trying to work out the $d = 2$ case in particular.
$$\mathcal C^d = \{(x\_1, \ldots, x\_{d+1}): |(x\_1, \ldots, x\_d)| = |x\_{d+1}|\} \subset \Bbb R^{d+1}$$
is the $d$-dimensional cone. The fir... | https://mathoverflow.net/users/157422 | $|\hat\mu(\xi)| \lesssim |\xi|^{-1/2}$ where $\mu$ is $f\mapsto \int_{\mathbb R} \psi(r) \int_{S^{1}} f(rx,r)\, d\sigma(x)\, dr $ | When the Hessian has rank $k$ everywhere one has an estimate $|\hat{\mu}(\xi)| \leq C|\xi|^{-{k \over 2}}$ by a result of Littman (see 5.8 on p 361 of Stein's Harmonic Analysis). Here $k = 1$ since the cone portion here has exactly one nonvanishing principal curvature at every point. The proof is basically the same as ... | 1 | https://mathoverflow.net/users/149955 | 425196 | 172,660 |
https://mathoverflow.net/questions/425179 | 5 | Let $u(x)$ be a harmonic polynomial in the unit ball $B\_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B\_r(0)\vert}\int\_{B\_r(0)}\vert\nabla u\vert^2dx,$$
and the average of the square function on the boundary
$$B(r):=\frac1{\vert \partia... | https://mathoverflow.net/users/66131 | A constant ratio of integrals? Part I | No. E.g., if $n=2$ and $u(x,y)=x + x^2 - y^2$ for $(x,y)\in\mathbb R^2$, then $\dfrac{r^2A(r)}{B(r)}=2\dfrac{1+2r^2}{1+r^2}$.
| 6 | https://mathoverflow.net/users/36721 | 425199 | 172,661 |
https://mathoverflow.net/questions/425203 | 4 | This question is a follow up on my latest [MO post](https://mathoverflow.net/questions/425179/a-constant-ratio-of-integrals-part-i) which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context required me to look into spherical har... | https://mathoverflow.net/users/66131 | A constant ratio of integrals? Part II | In fact, this is true for any homogeneous polynomial $u$ (not identically $0$), be $u$ harmonic or not.
Indeed, let $m\ge1$ be the degree of such a polynomial $u$. Then
$$u(tx)=t^m u(x)$$
and
$$v(tx)=t^{2m-2} v(x)$$
for all real $t$, where $v:=|\nabla u|^2$. So, for $B\_r:=B\_r(0)$,
$$\int\_{B\_r}v(x)\,dx =\int\_{B\_... | 5 | https://mathoverflow.net/users/36721 | 425207 | 172,663 |
https://mathoverflow.net/questions/425208 | 2 | Let $(X\_{n,k})\_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\frac1n \sum\_{k=1}^n X\_{n,k} \right] \xrightarrow{n\to\infty} \mu
$$
If we further know that
$$
\mathbb{E}\left[\lef... | https://mathoverflow.net/users/23959 | Law of large numbers for triangular arrays whose moments "look independent" | $\newcommand\ep\varepsilon$Let
$$Z\_n:=\frac1n \sum\_{k=1}^n X\_{n,k}. \tag{1}\label{1}$$
We have $EZ\_n\to\mu$ and $EZ\_n^2\to\mu^2$, whence $\operatorname{Var}\,Z\_n=EZ\_n^2-(EZ\_n)^2\to0$. So, for each real $\ep>0$, by Chebyshev's inequality,
$$P(|Z\_n-EZ\_n|>\ep)\le\frac{\operatorname{Var}\,Z\_n}{\ep^2}\to0.$$
So, ... | 2 | https://mathoverflow.net/users/36721 | 425211 | 172,664 |
https://mathoverflow.net/questions/425182 | 18 | Sometimes, a mathematician dies suddenly, leaving behind very good mathematics that didn't make it through the publication pipeline. For example, it is possible they had a paper entirely ready to submit (maybe even already shared with their network, or on arxiv). Or they might even have submitted the paper and then die... | https://mathoverflow.net/users/11540 | How to get a research paper published after the author has died? | The OP asks for a specific "step by step process to get a paper published by a deceased person who wrote the paper alone".
A statement on this was made a few years ago by [COPE](https://publicationethics.org/case/deceased-author) (Committee on Publication Ethics). This involved a case where the manuscript was submitt... | 11 | https://mathoverflow.net/users/11260 | 425227 | 172,670 |
https://mathoverflow.net/questions/425239 | 4 | In [1,2] the authors proved the *Hard Lefschetz theorem in intersection cohomology*:
>
> Let $Z$ be a complex projective variety of pure complex dimension $d$, with $\xi\in H^2(Z,\mathbb{Q})$ the first Chern class of an ample line bundle over $Z$. Then there are isomorphisms given by cup product
> $$
> \xi^k\cdot\\... | https://mathoverflow.net/users/57030 | Hard Lefschetz theorem in intersection cohomology | For an algebraically closed field, you need to use the étale topology as the analytic topology is not available. You then must use $\mathbb Q\_\ell$-coefficients rather than $\mathbb Q$-coefficients. As far as I know, you must use the formalism of the derived category - I don't know how to define intersection homology ... | 7 | https://mathoverflow.net/users/18060 | 425242 | 172,671 |
https://mathoverflow.net/questions/425217 | 30 | In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, thanks in part to [the solution of the Kervaire Invariant One Problem by Hill, Hopkins, and Ravenel](https://annals.math.... | https://mathoverflow.net/users/11540 | What happened to the last work Gaunce Lewis was doing when he died? | Lewis wrote, but never published, a very influential paper setting foundations for the multiplicative theory for Mackey functors. The paper is called "The Theory of Green functors" and [Doug Ravenel's paper archive has a scanned copy](https://people.math.rochester.edu/faculty/doug/otherpapers/Lewis-Green.pdf). The date... | 27 | https://mathoverflow.net/users/360 | 425254 | 172,674 |
https://mathoverflow.net/questions/425126 | 0 | On the page 6 of the paper [Simulating quantum circuits by contracting tensor network](https://arxiv.org/abs/quant-ph/0511069) author wrote about the equivalence of *treewidth* of a Graph and its *induced width* where the *treewidth* is defined through the tree decompositions and the *induced width* - as the maximum de... | https://mathoverflow.net/users/99204 | What is the proof of the graph treewidth and the induced graph width equivalence? | If a graph $G$ has treewidth $\leq k$, then it is a subgraph of a [$k$-tree](https://en.wikipedia.org/wiki/K-tree).
By taking a [perfect elimination sequence](https://en.wikipedia.org/wiki/Chordal_graph#Perfect_elimination_and_efficient_recognition), the graph $G$ turns our to have induced width $\leq k$.
Conversely,... | 1 | https://mathoverflow.net/users/125498 | 425255 | 172,675 |
https://mathoverflow.net/questions/425240 | 2 | I meet this problem when reading Rolfsen's Knots&Links. After giving 8 different definitions of linking number for knots in $S^3$, he left an exercise: Given disjoint PL knots $J$ and $K$ in $S^3=\partial D^4$, let $A$ and $B$ be 2-chains in $D^4$, such that $\partial A=J,\partial B=K$, assume $A$ and $B$ intersect tra... | https://mathoverflow.net/users/483712 | A definition of linking number for knots in $S^3$ using chains in $D^4$ | Here is a brief sketch. First, show that the intersection number between A and B is independent of the choice of specific chains. (In other words, if $A'$ and $B'$ are other 2-chains with the same properties, then $A \cdot B = A'\cdot B'$.)
Now you can compare this with one of the other definitions that take place in... | 3 | https://mathoverflow.net/users/3460 | 425261 | 172,678 |
https://mathoverflow.net/questions/425258 | 5 | Corollary 9 in [these notes](https://cmsa.fas.harvard.edu/wp-content/uploads/2022/03/immersions-revised2.pdf) by Ralph Cohen has grabbed my attention.
>
> I do not undestand how to show that if we have a rank $k$ bundle which is *stably* isomorphic to the stable normal bundle then there is a virtual normal bundle o... | https://mathoverflow.net/users/99042 | Stable normal bundle and immersions | This follows from obstruction theory; also see [this answer](https://mathoverflow.net/a/417820/21564).
If $E \to X$ is a rank $r$ real vector bundle over a CW complex $X$, then the obstructions to finding a nowhere-zero section lie in $H^i(X; \pi\_{i-1}(S^{r-1}))$. In particular, if $r > \dim X$, then such a section ... | 6 | https://mathoverflow.net/users/21564 | 425262 | 172,679 |
https://mathoverflow.net/questions/424932 | 4 | Let $K$ be a local field of characteristic zero and $X$ an affinoid rigid space over $K$. Let $U\subset X$ be an affinoid subdomain, and consider a finite family of points $\{p\_{1},\cdots, p\_{n}\}\subset U$. Is it true that there is a rational subdomain $V\subset X$ such that $\{p\_{1},\cdots, p\_{n}\}\subset V\subse... | https://mathoverflow.net/users/476832 | On a consequence of the Gerritzen-Grauert Theorem | Yes, this is true . You can proceed as follows. First pick finite many functions $f\_1,\dots,f\_m$ on $U$ such that $V(f\_1,\dots,f\_m)$ consists of finitely many points, including the ones you want. (For instance, if you embed $U$ into a polydisk, you can take a non-constant polynomial $P\_i$ that vanishes on the $i$-... | 2 | https://mathoverflow.net/users/4069 | 425265 | 172,680 |
https://mathoverflow.net/questions/425267 | 5 | let $\xi,\eta: \Omega \to \mathbb R$ be i.i.d. random variables on a measurable space $(\Omega , \mathcal F,\mathbb P)$, and let $f: \mathbb R^2 \to \mathbb R$ be a bivariate measurable function (say under Borel $\sigma$-algebra). Clearly, $\omega \mapsto f(\xi(\omega),\eta(\omega))$ is also a random variable.
In man... | https://mathoverflow.net/users/484625 | Why is it valid to take uncountable infimum of one dimension of a multivariate function of random variables? | $\newcommand\si\sigma\newcommand\om\omega\newcommand\Om\Omega\newcommand\R{\mathbb R}\newcommand\F{\mathcal F}\newcommand\B{\mathcal B}$No, $g$ is not in general Borel measurable, even if $f$ is Borel measurable.
E.g., let $\Om:=\R$ with $\F:=\B(\R)$, the Borel $\si$-algebra over $\R$. Let $\eta(\om):=\om$ for all $\... | 6 | https://mathoverflow.net/users/36721 | 425272 | 172,683 |
https://mathoverflow.net/questions/425147 | 1 | I am looking for a reference with a proof for the following fact:
If a right-continuous martingale $(X\_r)\_{ r \geq 0}$ is such that $X\_0=0,(X^2\_r-r)\_r,(X\_r^3-3rX\_r)\_r,(X\_r^4-6rX\_r^2+3r^2)\_r$ are martingales then $(X\_r)\_{r}$ is a Brownian motion.
| https://mathoverflow.net/users/138491 | Characterization of Brownian motion: processes with right-continuous paths | It suffices to prove that the paths are continuous, and then the result follows from
<https://almostsuremath.com/2010/04/13/levys-characterization-of-brownian-motion/>
A version of this question was also asked on math stack exchange, for convenience I include the proof also here.
**Proposition**
Let $(X\_u)\_{u}$ b... | 1 | https://mathoverflow.net/users/7691 | 425276 | 172,687 |
https://mathoverflow.net/questions/424966 | 1 | Let $p \equiv 1 \pmod{3}$ be a prime and denote $H\_{n,m} = \sum\_{k = 1}^n 1/k^m$ as the $n,m$-th generalized harmonic number. I'm interested in computing $H\_{(p-1)/3,\, 2}$ and $H\_{(p-1)/6,\,2}$ modulo $p$. From [this](https://www.jstor.org/stable/1968791) paper I know
\begin{align\*}
H\_{(p-1)/6,2} \equiv -\frac{B... | https://mathoverflow.net/users/171396 | A question about generalized harmonic numbers modulo $p$ | Glaisher's I-numbers are described in J. W. L. Glaisher, [On a set of coefficients analogous to the Eulerian numbers](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s1-31.1.216), Proc. London Math. Soc., 31 (1899), 216-235.
| 3 | https://mathoverflow.net/users/10744 | 425284 | 172,689 |
https://mathoverflow.net/questions/425285 | 4 | The Bockstein SS is obtained from the exact sequence
$$0\to\mathbb{Z}\xrightarrow{2}\mathbb{Z}\to\mathbb{Z}/2\to 0$$
with $E\_1^p=H^p(X,\mathbb{Z}/2)$ and the differential $d\_1=Sq^1$.
How to identify the differentials $d\_2$ for the $E\_2$-page without knowing $H^\*(X,\mathbb{Z})$ in advance?
| https://mathoverflow.net/users/149491 | Higher order differentials of Bockstein spectral sequence | The $E\_1$ page does not tell you what the higher differentials will be, and you will have to know at least something about the integral cohomology. Consider the case when $X$ is a Moore space $M(1,\mathbb{Z}/2^n)$ which you may prefer to think of as a circle with a disc attached via a $\times 2^n$ map. In each case th... | 6 | https://mathoverflow.net/users/124004 | 425305 | 172,694 |
https://mathoverflow.net/questions/425300 | 2 | Let $A,D \in \mathbb{R}^{n\times n}$ be two positive definite matrices given by
$$
D =
\begin{bmatrix}
1 & -1 & 0 & 0 & \dots & 0\\
-1 & 2 & -1 & 0 & \dots & 0\\
0 & -1 & 2 & -1 & \dots & 0\\
\vdots & \ddots & \ddots & \ddots & \ddots & 0\\
0 & \dots & 0 & -1 & 2 & -1\\
0 & 0 & \dots & 0 & -1 & 1
\end{bmatrix}, \quad... | https://mathoverflow.net/users/484658 | Prove spectral equivalence of matrices | For $i=1,\dots,n-1$, let $a\_i:=c\_+-c\_{i,i+1}\ge0$. Then, by straightforward calculations with a bit of re-arranging, for $x=(x\_1,\dots,x\_n)\in\mathbb R^n$ we have
$$c\_+ x^\top D x-x^\top A x
=\sum\_{i=1}^{n-1}a\_i(x\_{i+1}-x\_i)^2\ge0.$$
So, your conjectured inequality, $x^\top A x \le c\_+ x^\top D x$, is true... | 2 | https://mathoverflow.net/users/36721 | 425312 | 172,696 |
https://mathoverflow.net/questions/425324 | 1 | Consider the drift Brownian motion $X\_t:=1+bt+W\_t$, where $(W\_t)\_{t\ge 0}$ is a Brownian motion starting at zero. Set $\tau:=\inf\{t\ge 0: X\_t=0\}$. Assume $b>0$, then $\mathbb P[\tau=\infty]>0$. What is the conditional law of $X\_{\infty}$ knowing $\tau=\infty$?
| https://mathoverflow.net/users/261243 | Conditional probability distribution of a Brownian particle surviving forever | By the law of Large numbers, $X\_t/t \to b$ almost surely as $t \to +\infty$, hence $X\_t \to +\infty$ almost surely as $t \to +\infty$. Therefore $X\_\infty = +\infty$ almost surely under $P$ and also under $P[\cdot|\tau = \infty]$.
| 2 | https://mathoverflow.net/users/169474 | 425325 | 172,699 |
https://mathoverflow.net/questions/425152 | -1 | This is a cross post in continuation to [this question](https://math.stackexchange.com/q/4472299/) on Mathematics Stack Exchange. I wanted to know if this inequality holds true in two or three dimensions,
$\|\nabla\phi\|\_{L^{\infty}(\Omega)}\leq C\|\phi\|\_{H^2(\Omega)}.$
Where $\Omega$ is an open-bounded domain a... | https://mathoverflow.net/users/131281 | Sobolev estimates $\|\nabla\phi\|_{\infty}\leq C\|\phi\|_{H^2}$ | In $d \geq 3$ the answer is no from scaling argument.
WLOG we can assume $0\in \Omega$ (by translation) and that $B(0,r\_0)\subset\Omega$. Take $\phi\in C^\infty\_0(B(0,r\_0))\subset C^\infty\_0(\Omega)$. Define
$$ \phi\_{\lambda}(x) = \lambda^{2 - d/2}\phi(x/\lambda) $$
Note that when $\lambda \in (0,1)$ the fun... | 1 | https://mathoverflow.net/users/3948 | 425327 | 172,701 |
https://mathoverflow.net/questions/425297 | 2 | Let $\phi$ be a continuous function on the closed upper half-plane $\{ z\in\mathbb{C}: \operatorname{Im}(z)\ge 0\}$ and holomorphic in the interior.
Suppose that the function $x\phi(x)$ is in $C^1(\mathbb{R})$.
Does it follow that $\phi$ is in $C^1(\mathbb{R})$, too?
Without the holomorphy, this is false, but maybe h... | https://mathoverflow.net/users/473423 | Singularity on the boundary of domain of holomorphy | Write $\phi=\phi\_1+i\phi\_2$. A counterexample is given by
$$
\phi\_2(x)=\begin{cases} 0 & x<0 \\ x & 0\le x\le 1 \end{cases} .
$$
We also give $\phi\_2$ compact support and keep it smooth away from $x=0$.
We can then set
$$
\phi(z) = \frac{1}{\pi}\int\_{-\infty}^{\infty} \frac{\phi\_2(t)\, dt}{t-z}\label{2}\tag{2}
$$... | 2 | https://mathoverflow.net/users/48839 | 425337 | 172,704 |
https://mathoverflow.net/questions/425334 | 13 | In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these [notes](https://www.msri.org/workshops/918/schedules/28233/documents/50487/assets/88599) or this [lecture](https://youtu.be/pixkOp_KK-M?t=5499)) some important aspects of the Langlands correspondence are stated in the language of topological qu... | https://mathoverflow.net/users/85392 | Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh | A curve $C$ over $\mathbb F\_q$ has dimension $3$ in this perspective (which is why you get a vector space) and a local field has dimension $2$ (which is why you get a category. So one only has to go up one dimension higher to get the four-dimensional theory.
Associated to the "product $C \times S^1$", we would take ... | 11 | https://mathoverflow.net/users/18060 | 425338 | 172,705 |
https://mathoverflow.net/questions/425341 | 4 | Let $f : \mathbb{R} \to \mathbb{R}$ be a function of normalized bounded variation (NBV), meaning that $f$ is of bounded variation, $f$ is right continuous, and $f(x) \to 0$ and $x \to -\infty$. As explained in Section 3.5 of Folland's Real Analysis textbook, there is a unique complex measure $df$ with the property that... | https://mathoverflow.net/users/87862 | Chain rule for $e^f$, where $f$ has bounded variation | $\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}\newcommand{\ep}{\varepsilon}$No reasonable chain rule will hold here in general, if $f$ is allowed to be discontinuous.
Indeed, let $\mu\_f:=df$, the Lebesgue--Stieltjes measure corresponding to the right-continuous function $f$ of bounded variation. Similarly defin... | 3 | https://mathoverflow.net/users/36721 | 425349 | 172,711 |
https://mathoverflow.net/questions/425354 | 13 | A Latin square of order $n$ has $n$ broken diagonals and $n$ broken antidiagonals. When $n \equiv \pm 1 \pmod 6$, we have *diagonally cyclic Latin squares* in which those $2n$ diagonals are transversals (i.e., every symbol occurs exactly once). For example
$$
\begin{bmatrix}
4 & \color{red} 3 & 2 & 1 & \color{blue} 0... | https://mathoverflow.net/users/48278 | Does there exist a Latin square of order 9 for which its 9 broken diagonals and 9 broken antidiagonals are transversals? | I've verified with ILP that such Latin squares do not exist for $n\in\{9,15,21,27\}$.
The ILP formulation is based on binary indicator variables $p\_{c,i,j}$ telling whether Latin square has character $c$ at position $(i,j)$, with constraints
$$\begin{cases}
\sum\_i p\_{c,i,j} = 1, \\
\sum\_j p\_{c,i,j} = 1, \\
\sum\... | 14 | https://mathoverflow.net/users/7076 | 425357 | 172,712 |
https://mathoverflow.net/questions/425359 | 3 | I have a $C^\*$-algebra $\mathcal{A}$, and would like to make use of the spectral order $\preceq$ coming from (the self-adjoint part of) its enveloping von Neumann algebra $\mathcal{A}^{\*\*}$.
I am most interested in checking that the spectral join/meet of finite subsets of $\mathcal{A}^\text{sa}$ are contained in $... | https://mathoverflow.net/users/480800 | Spectral join in a $C^*$-algebra relative to its enveloping von Neumann algebra | Sorry, the answer is no. In general the spectral join of two positive operators $a$ and $b$ is the strong operator limit of $(a^k + b^k)^{1/k}$ as $k \to \infty$ (Corollary 10 of [this](https://www.ams.org/journals/proc/1996-124-11/S0002-9939-96-03474-0/S0002-9939-96-03474-0.pdf) old old paper of mine). But this need n... | 4 | https://mathoverflow.net/users/23141 | 425362 | 172,714 |
https://mathoverflow.net/questions/425369 | 2 | Let $W=\{W\_t\}\_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F\_t\}\_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma\_t\}\_{t\in[0;1]}$ be a continuous and bounded Ito process (**EDIT**: w.r.t. $W$) with bounded drift and volatility coefficients (I could also live with mo... | https://mathoverflow.net/users/130906 | Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral | This edit reflects the actual question asked, and corrects an earlier answer.
You can rewrite the process $\int\_0^t \sigma\_s dW\_s$ as a time change of Brownian motion, where the time change is given by $\tau(t)=\int\_0^{t} \sigma^2(s) ds$.
If $\sup\_{t\in [0,1]} |\sigma\_t|<R$ a.s. for some deterministic $R$, then... | 2 | https://mathoverflow.net/users/35520 | 425370 | 172,716 |
https://mathoverflow.net/questions/425375 | 3 | Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A x \le c^+ x^\top D x $$
for any $x \in \mathbb{R}^n$, where $c^+, c^- > 0.$ The matrices $A$ and $D$ can be diagonalized... | https://mathoverflow.net/users/484661 | prove spectral equivalence bounds for fractional power of matrices | $\newcommand\C{\mathbb C}\newcommand\R{\mathbb R}\newcommand\al{\alpha}$Yes, this follows by Loewner's theorem on monotone matrix functions (see e.g. [Theorem 1.6](https://link.springer.com/book/10.1007/978-3-030-22422-6)), which in particular implies the following:
Let $M\_n$ denote the set of all analytic functions... | 2 | https://mathoverflow.net/users/36721 | 425379 | 172,718 |
https://mathoverflow.net/questions/425378 | 1 | This question has been migrated from the [MSE](https://math.stackexchange.com/questions/4478972/estimating-the-variance-of-monte-carlo-estimators-for-f-z-and-f-z-z-x-y).
**Background/Motivation:**
We have $Z=X/Y$ where $X$ and $Y$ are independent and $X\sim\mathcal N(\mu,\sigma^2)$. The density of $Y$ is not import... | https://mathoverflow.net/users/125801 | Estimating the variance of Monte Carlo estimators for $F_Z$ and $f_Z$, $Z=X/Y$ | $\newcommand{\de}{\delta}\newcommand{\si}{\sigma}$Of course, $\de^2$ makes no sense. So, do not use $\de$.
Instead, write
\begin{equation}
f\_Z(z)=E g(Y)\approx \hat f\_Z(z):=\frac{1}{n}\sum\_{k=1}^n g(Y\_k),
\end{equation}
where
\begin{equation}
g(y):=\frac{|y|}\si\,\phi\Big(\frac{zy-\mu}\si\Big).
\end{equation}
... | 1 | https://mathoverflow.net/users/36721 | 425383 | 172,719 |
https://mathoverflow.net/questions/425380 | 2 | Let $v\in \mathbb{R}^d$ be a random vector such that $\mathbb{E}v = y$, and $X$ be a given $\mathbb{R}^{d\times d}$ real fixed matrix. We assume that the following optimization problem has a unique solution $w^\star$:
$$\min\_{w\in \mathcal{C}}f(w):=\sum\_{i=1}^d |y\_i-(Xw)\_i|^2$$ where $\mathcal{C}$ might be a convex... | https://mathoverflow.net/users/64194 | On the relation between solution of random least squares and expected least squares with constraints | $\newcommand\C{\mathcal C}$If $\C$ is convex, then the projection onto the convex set $X\C$ is [uniquely defined and, moreover, $1$-Lipschitz](https://math.stackexchange.com/questions/3272169/projections-onto-convex-sets-and-lipschitz-condition), so that
$\|X\hat w-Xw^\ast\|\le\|v-Ev\|$ and hence $\|\hat w-w^\ast\|\le\... | 1 | https://mathoverflow.net/users/36721 | 425388 | 172,720 |
https://mathoverflow.net/questions/425387 | 1 | For $x\_+ \in (0,\infty)$ let $f\colon(0,x\_+] \to (0,\infty)$ be a continous differentiable function with $f(x) > 0$ and $f'(x) < 0$ for all $x \in (0,x\_+]$.
Moreover, we assume that
$$\lim\_{x \to 0} f(x) = \infty$$
holds.
**The question:** Does this implies that there exists a $\beta \in (0,\infty)$ such that $f(... | https://mathoverflow.net/users/484700 | Positive, monotone decreasing function, with limit in 0 equal to ∞ submultiplicative up to an factor? | A counterexample is given by
$$f(x)=e^{1/x}.$$
Indeed, then $f(x)f(x)/f(xx)\to0$ as $x\downarrow0$, so that, for any real $x\_+$, there is no real $\beta>0$ such that $f(x)f(y)\ge\beta f(xy)$ for all $x,y$ in $(0,x\_+]$
| 2 | https://mathoverflow.net/users/36721 | 425390 | 172,721 |
https://mathoverflow.net/questions/425400 | 3 | Consider the following integral expression:
$$\mathcal I :=\iint\_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$
for $\epsilon>0$, $f \in L^\infty(\mathbb R)$, and $g \in BV(\mathbb R)$.
Is it true that
$$\mathcal I \lesssim TV(g)$$
or something of this nature (possibly adding the... | https://mathoverflow.net/users/nan | Bounding integral expression with total variation of integrand | $\newcommand{\ep}{\epsilon}\newcommand{\R}{\mathbb R}$Yes, this is true. Indeed, for $\ep\in(0,1/2]$ we have
\begin{equation}
\mathcal I\le2\|f\|\_\infty^2\, J, \tag{1}\label{1}
\end{equation}
where
\begin{equation}
\begin{aligned}
J&:=\iint\_{\R^2}\,\frac{dx\, dy}{(x-y)^2}\,|g(x)-g(y)|\,1(\ep\le y-x\le1/2) \\
&\le... | 4 | https://mathoverflow.net/users/36721 | 425408 | 172,724 |
https://mathoverflow.net/questions/425336 | 4 | This question regards a part of the proof of the so called *surgery step*, in Wall's [book](https://www.maths.ed.ac.uk/%7Ev1ranick/books/scm.pdf) "surgery on compact manifolds", Theorem 1.1.
**Setting**
$M^m$ smooth manifold, $X$ CW complex, $\phi :M\to X$ continuous map, $\nu\to X$ a rank-$v$ vector bundle and $F:... | https://mathoverflow.net/users/99042 | On the proof of the surgery step in Wall's book | The theorem has the hypothesis "$f$ is in this class", meaning that the embedding $f$ is in the regular homotopy class of immersions determined by $F$ together with the element of the relative homotopy group $\pi\_{r+1}(\phi)$ that is given by the maps $f\_0$ and $Q$. This hypothesis actually says that the two stable t... | 4 | https://mathoverflow.net/users/6666 | 425413 | 172,726 |
https://mathoverflow.net/questions/425371 | 8 | Let $ A$ be a complex $ n$ by $ n$ matrix and $ x\_1, \dots, x\_n$ be a set of commuting variables. Let $ X\_i = \sum\_i a\_{ij}x\_j$. MacMahon's Master Theorem (MMT) states that
\begin{align}
[x\_1^{p\_1} \dots x\_n^{p\_n}] X\_1^{p\_1} \dots X\_n^{p\_n} = [s\_1^{p\_1} \dots s\_n^{p\_n}] \det(I- SA)^{-1}
\end{align}
w... | https://mathoverflow.net/users/97209 | MacMahon Master Theorem for non-matching coefficients | The reason I asked this question is that I found such a generalization of MMT and didn't know if it exists in the literature. The proof makes extensive use of **operator calculus**: the differential operators are manipulated as if they were numbers. The MMT and its generalization are immediate consequences of the follo... | 9 | https://mathoverflow.net/users/97209 | 425429 | 172,732 |
https://mathoverflow.net/questions/425437 | 0 | Higher inductive types are a useful concept in homotopy type theory. However, considering its general syntax is a bit of a challenge. Is it possible to implement all higher inductive types with just generalized algebraic data types and [non-truncated quotients](https://github.com/agda/cubical/blob/5c22f5dcaaddaeb7f07df... | https://mathoverflow.net/users/150063 | Construct higher inductive types with only generalized algebraic data types and non-truncated quotients? | No, it is not.
One can do a lot with just homotopy pushouts/coequalizers (I assume this is what you mean by "non-truncated quotients"). For instance, Egbert Rijke showed in [The join construction](https://arxiv.org/abs/1701.07538) that from this together with the natural numbers you can construct truncations.
Howev... | 6 | https://mathoverflow.net/users/49 | 425454 | 172,736 |
https://mathoverflow.net/questions/425397 | 4 | Let $\cal X$ be a DM stack and ${\cal D}\hookrightarrow{\cal X}$ an effective Cartier divisor on it. Suppose that $n$ is a positive integer invertible in ${\cal X}$.
Let $\sqrt[n]{{\cal D}}\to{\cal X}$ be the root stack associated to the triple $({\cal X, D}, n)$.
(1) Is it true that the moduli space of $\sqrt[n]{{... | https://mathoverflow.net/users/116681 | Questions about root stacks | One way to construct the root stack is to consider the universal situation of $\Theta := [\mathbb{A}^1/\mathbb{G}\_m]$. Here let us work over $\mathbb{Z}[\frac{1}{n}]$ since we assume $n$ is invertible. There is a natural map $\phi\_n : \Theta \to \Theta$ induced by $z \mapsto z^n$. Given an effective Cartier divisor $... | 7 | https://mathoverflow.net/users/12402 | 425460 | 172,740 |
https://mathoverflow.net/questions/425447 | 0 | Let $X\_1$ be the suspension of $\mathbb{R}P^2$ and $X\_2=\bigvee\_{1\leq i\leq n} (\vee\_{r\_i} \mathbb{S}^i)$.
Is $\pi\_2 (X\_i)$ a projective (or a free) $\mathbb{Z}\pi\_1 (X\_i)$-module for $i=1,2$?
I was wondering if someone could help me about my question. I don't have much information about $\pi\_... | https://mathoverflow.net/users/114476 | Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$? | No and yes, respectively.
For $X\_1$, the suspension is simply connected, so $\pi\_2(X\_1) = H\_2(X\_1) = H\_1(\mathbb{R}P^2) = \mathbb{Z}/2$ which is not a free $\mathbb{Z}$ module.
For the other one, $\pi\_2(X\_2) = H\_2(\tilde{X}\_2)$ where $\tilde{X}\_2$ is the universal cover. Since $\pi\_1(X\_2)$ is a free gr... | 2 | https://mathoverflow.net/users/3460 | 425463 | 172,742 |
https://mathoverflow.net/questions/425430 | 0 | Related to [this](https://mathoverflow.net/questions/425387/positive-monotone-decreasing-function-with-limit-in-0-equal-to-%e2%88%9e-submultiplica) question.
For $x\_+ \in (0,\infty)$, $a \in \mathbb{R}$ let $F\colon[0,x\_+] \to [a,\infty)$ be a twice continuous differentiable (in $(0,x\_+)$) function with $f := F'$,... | https://mathoverflow.net/users/484700 | Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicative up to an factor? | Define $f$ on $]0,e^{-2}]$ by
$$f(x) = \frac{1}{-\sqrt{x}\ln(x)}.$$
Then $f$ is positive, decreasing since
$$\frac{\mathrm{d}}{\mathrm{d}x}\big(-\sqrt{x}\ln(x)\big) = \frac{-\ln(x)-2}{2\sqrt{x}} \ge 0 \text{ for all } x \in (0,e^{-2}].$$
Moreover $f(x) \to +\infty$ as $x \to 0$. But since $f(x) = o(x^{-2/3})$ as $x \to... | 2 | https://mathoverflow.net/users/169474 | 425465 | 172,743 |
https://mathoverflow.net/questions/425449 | 5 | Let $D\_n$ be the set of divisors of $n$.
Does there always exists a $B\subseteq D\_n$ such that $D\_n = \{\gcd(ab,n) \mid a\leq \sqrt{n}, b\in B\}$ and $\sum\_{b\in B} \frac{n}{b}=O(n)$?
| https://mathoverflow.net/users/6886 | Small covering of divisors | **UPDATED**
We can simply take
$$B = \{1\} \cup \{ d\in D\_n\ :\ d > n^{1/2} \}.$$
Then for any $d\in D\_n$:
* if $d\leq n^{1/2}$, we take $(a,b)=(d,1)$;
* if $d>n^{1/2}$, we take $(a,b)=(1,d)$.
Then
$$\sum\_{b\in B} \frac{n}{b} = n + O(n^{1/2}\cdot\tau(n)) = O(n)$$
as required.
| 7 | https://mathoverflow.net/users/7076 | 425467 | 172,744 |
https://mathoverflow.net/questions/425358 | 4 | Let $V^{m+1} = \mathbb{C}^{m+1}$ and let $U(1)$ act on it by its diagonal representation, so that really, it is just like scalar multiplication by a unit modulus complex number.
I am interested in the quotient $Q^m = V^{m+1}/U(1)$, which has real dimension $2m+1$. What I am interested in is the following question. Wh... | https://mathoverflow.net/users/81645 | Is this quotient of $\mathbb{C}^{m+1}$ by $U(1)$ only "nice" for $m=1$? | The quotient is a cone on $\mathbb{CP}^m$.
When $m$ is even, $\mathbb{CP}^m$ is not the boundary of any compact smooth $(2m{+}1)$-manifold, so you can't smooth the singularity at the tip of the cone by blowing-up or any similiar local modification.
When $m$ is odd, $\mathbb{CP}^m$ is the boundary of a $3$-disk bund... | 5 | https://mathoverflow.net/users/13972 | 425471 | 172,746 |
https://mathoverflow.net/questions/425406 | 4 | I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this prompted me to wonder about the possibility of a similar sort of story for harmonic functions and the Dirichlet problem.
... | https://mathoverflow.net/users/419791 | Harmonic functions as limits of harmonic functions on graphs? | Such approximations have a long history, starting with
[1] Courant, R., Friedrichs, K. and Lewy, H. (1928) Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 32–74
Good error estimates are in
[2] Laasonen, P. (1967) On the discretization error of the Dirichlet problem in a plane ... | 7 | https://mathoverflow.net/users/7691 | 425480 | 172,753 |
https://mathoverflow.net/questions/425439 | 4 | On an $(M,g)$ compact n-dimensional Riemannian manifold the Yamabe operator $Y = 4(n-1)/(n-2)L + s$, where $L$ is the Laplacian and $s$ is the scalar curvature is conformally covariant, which for me just means that transforms in a simple way under conformal transformations, $g \to f g$ then $Y \to f^\alpha Y f^\beta$ w... | https://mathoverflow.net/users/41312 | Yamabe operator, conformal transformations and square of the Dirac operator | The discrepancy comes from the fact that the square of the Dirac operator is in general not conformally covariant. There are ways to modify powers of the Dirac operator to get a conformally covariant operator (see, for example, [this article](https://arxiv.org/abs/1311.4182) of Fischmann). However, they involve adding ... | 3 | https://mathoverflow.net/users/121820 | 425504 | 172,758 |
https://mathoverflow.net/questions/425495 | 0 | Consider the following integral expression:
$$\mathcal I :=\iint\_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\infty(\mathbb R^n;\mathbb R)$ and *$f$ compactly supported* (NB compact support assumption added later).
If $g \in W^{1,p}(... | https://mathoverflow.net/users/nan | Bounding integral expression with Sobolev norm of integrand | Such a bound is impossible already for $n=1$. Indeed, suppose that $f=1$ and
$$g'(x)=\frac1{1+|x|}$$
for all real $x$. Then $TV(g)=\infty$ and hence, according to the [previous answer](https://mathoverflow.net/a/425408/36721), $\mathcal I=\infty$, whereas $\|\nabla g\|\_{L^p(\mathbb R^n)}<\infty$ for any $p\in(1,\infty... | 1 | https://mathoverflow.net/users/36721 | 425514 | 172,762 |
https://mathoverflow.net/questions/425512 | 3 | Consider the all-familiar *Vandermonde determinant* $V\_n(x\_1,\dots,x\_n)$ of the matrix of $(i,j)$-entries $M\_n(i,j)=x\_j^{i-1}$ so that
$$V\_n(x\_1,\dots,x\_n)=\prod\_{1\leq i<j\leq n}(x\_j-x\_i).$$
Let's specialize the variables $x\_k=k\pi\_k$ where $\pi\in\mathfrak{S}\_n$ is a permutation of $n$ letters $\{1,2,\d... | https://mathoverflow.net/users/66131 | Vandermonde $V_n$ mod $n$ | Per comments above, for a counterexample we have with necessity $\pi\_n=n$ and prime $n$. The case $n=2$ is trivial, so I assume that $n$ is an odd prime.
The elements $U:=\{ 1,2,\dots,n-1\}$ form the unit group of $GF(n)$ and the mapping $i\mapsto i\pi\_i$ has to be a permutation of $U$. Such mappings are called *co... | 8 | https://mathoverflow.net/users/7076 | 425518 | 172,764 |
https://mathoverflow.net/questions/425419 | 7 | What is exactly demanded for a set theoretic foundation of Category theory? I saw generally two main approaches. One is [Muller](http://philsci-archive.pitt.edu/1372/1/SetClassCat.PDF)'s who did the work in Ackermann's set theory, but his criteria seem to hint at existence of a universe of $\sf ZFC$ and an $n$-iterativ... | https://mathoverflow.net/users/95347 | Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles? | The title asks:
>
> Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles?
>
>
>
The first line in the main body of the question is:
>
> What is exactly demanded for a set theoretic foundation of Category theory?
>
>
>
These are two different questions. As I men... | 5 | https://mathoverflow.net/users/3106 | 425527 | 172,767 |
https://mathoverflow.net/questions/425494 | 15 | In Theorem 2 of [these notes](https://cmsa.fas.harvard.edu/wp-content/uploads/2022/03/immersions-revised2.pdf), Ralph Cohen reformulates the main theorem of Hirsch-Smale theory merely in terms of normal bundles.
In particular, he says that if $N, M$ are two manifolds, $\dim N< \dim M$ then two immersions
$f\_1, f\_2:N\... | https://mathoverflow.net/users/99042 | Possible mistake in Cohen notes "Immersions of manifolds and homotopy theory" (version 27 March 2022) | The statement is false in two ways. First, two immersions might not even be homotopic even though their normal bundles are both, say, trivial. Second, even if $M=\mathbb R^m$, regular homotopy classes of immersions of $N$ correspond to homotopy classes of vector bundle monomorphisms from $TN$ to the trivial rank $m$ bu... | 14 | https://mathoverflow.net/users/6666 | 425528 | 172,768 |
https://mathoverflow.net/questions/261195 | 9 | A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology
it suffices to verify it for Zariski squares and the empty cover of the empty scheme.
Similarly, homotopy descent for the Nisnevich topology boils down to Nisnevich squares and the empty cover.
There is ... | https://mathoverflow.net/users/402 | Reference for the Brown-Gersten property for smooth manifolds | I typed up a proof of this result:
>
> [Numerable open covers and representability of topological stacks](https://arxiv.org/abs/2203.03120).
>
>
>
The result is proved in greater generaility for arbitrary numerable open covers of topological spaces, together with some applications, including criteria for repre... | 2 | https://mathoverflow.net/users/402 | 425534 | 172,770 |
https://mathoverflow.net/questions/425404 | 2 | The Problem
===========
Given $i, n, y$, I am trying to find a closed form solution for $a$ in the equation
$$
\sum\_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j}
=
0
$$
Do you have any recommendations on how to solve this problem?
Background/Motivation
=====================
Let me explain. I am trying to compute t... | https://mathoverflow.net/users/484742 | How to solve for $a$ in $\sum_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j} = 0$ | $$\sum\_{j=i}^n (a -j) \binom{n}{j} y^j (1-y)^{n-j}
=0$$
$$\Rightarrow a=i-\frac{y \binom{n}{i+1} \, \_2F\_1\left(2,i-n+1;i+2;\frac{y}{y-1}\right)}{(y-1) \binom{n}{i} \, \_2F\_1\left(1,i-n;i+1;\frac{y}{y-1}\right)}.$$
Check that for $i=1$, this simplifies to $a=n y \left(\frac{1}{(1-y)^{-n}-1}+1\right)$, which is the c... | 0 | https://mathoverflow.net/users/11260 | 425535 | 172,771 |
https://mathoverflow.net/questions/425525 | 6 | Let $R$ be a commutative ring. It is well-known that if $b \in R$ and $c \in R$ are two nilpotent elements with $b^k = 0$ and $c^\ell = 0$ (where $k$ and $\ell$ are positive integers), then $b+c$ is nilpotent again with $\left(b+c\right)^{k+\ell-1} = 0$.
I'm wondering if this has a converse of the following form:
... | https://mathoverflow.net/users/2530 | Splitting a nilpotent into square-zeros by ring extension | I can give a positive answer to question 1 and an answer to question 2.
Theorem: Let $R$ be a commutative algebra. Let $k$ and $\ell$ be two positive integers. Let $a\in R$ satisfy $a^{k+\ell-1}=0$. If $\binom{k+\ell-2}{k-1}$ is not a zero divisor in $R$, then there exists a commutative ring $S$ such that $R$ is a su... | 7 | https://mathoverflow.net/users/18060 | 425536 | 172,772 |
https://mathoverflow.net/questions/425539 | 4 | The question is an extention to the answered question [prove spectral equivalence bounds for fractional power of matrices](https://mathoverflow.net/questions/425375/prove-spectral-equivalence-bounds-for-fractional-power-of-matrices).
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-di... | https://mathoverflow.net/users/484661 | prove spectral equivalence bounds for inverse fractional power of matrices | By [Remark 1 after Theorem 4.1](https://link.springer.com/book/10.1007/978-3-030-22422-6), the matrix expression $-A^{-\alpha}$ is Loewner-nondecreasing in positive definite matrix $A$ if and only $0\le\alpha\le1$.
So, the inequalities
$$\frac{1}{(c^+)^\alpha} x^\top D^{-\alpha} x \le x^\top A^{-\alpha} x \le \frac{1... | 6 | https://mathoverflow.net/users/36721 | 425540 | 172,774 |
https://mathoverflow.net/questions/425558 | 12 | In [Kaniuth, Taylor, Induced representations of locally compact groups](https://www.zbmath.org/?q=an%3A1263.22005) on pages 9-10 it's claimed that if $G$ is a locally compact group with closed subgroups $N,H$, with $N$ normal in $G$, with $N\cap H=\{e\}$, and with $NH=G$, then $G$ is a topological semidirect product of... | https://mathoverflow.net/users/406 | Topological semi-direct products of groups | Here is a counterexample:
let $K$ be an infinite compact group, and $K^\delta$ be $K$ with the discrete topology.
Let $G$ be $K^\delta\times K$, let $N$ be equal to $K^\delta\times\{0\}$ and let $H$ be the diagonal. Then both $N,H$ are discrete, $G=NH$, $N\cap H=\{1\}$. But the canonical continuous group isomorphis... | 20 | https://mathoverflow.net/users/14094 | 425559 | 172,781 |
https://mathoverflow.net/questions/425571 | 3 | In this post I consider the equation $$k\cdot x=y^2+z^2(x^2-2)-2\tag{1}$$
over odd integers $y\geq 1$ and $z\geq 1$, and over integers $k\geq 1$ and very large Mersenne exponents $x$ such that $x^2-2$ is a prime number.
Previous Diophantine equation $(1)$ is consequence of Fermat's little theorem applied to the dioph... | https://mathoverflow.net/users/142929 | A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent | The equation (1) for a fixed $x$ is equivalent to the congruence:
$$y^2 \equiv 2(1+z^2)\pmod{x}.$$
For $x=25964951$, we have $2\equiv 3328351^2\pmod{x}$, and thus all solutions are obtained from those $z$ for which $1+z^2$ is a square modulo $x$ (there are $\frac{x-1}2$ such residue classes). Having such a $z$, we ob... | 3 | https://mathoverflow.net/users/7076 | 425577 | 172,784 |
https://mathoverflow.net/questions/425517 | 1 | Let $X$ be an integral scheme over a field. Let $G$ be a finite group acting on $X$ faithfully. Assume the quotient stack $[X/G]$ is separated (e.g., when $G$ acts on $X$ properly). Then $[X/G]$ is a separated Deligne-Mumford (DM) stack and there is a coarse moduli space
$$\pi:[X/G] \to X/G.$$
Is $\pi$ always a biratio... | https://mathoverflow.net/users/146366 | Birational morphisms from DM stacks to their coarse moduli spaces | Yes. For each nontrivial element $g\in G$, the fixed points form a closed set, which must not contain the whole space as then $g$ would act trivially (by reducedness). The complementary open set thus contains the generic point.
The (nonempty, by irreducibility) intersection of these open sets over all nontrivial $g\i... | 1 | https://mathoverflow.net/users/18060 | 425584 | 172,787 |
https://mathoverflow.net/questions/425550 | 1 | What is $\left\| f \right\|\_{ H\_{0}^{k}, H\_{0}^{k}}$ norm when $H\_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int\_{M} u \operatorname{vol}\_{g}=0\right\}$.
I'm reading a paper
[Chern-Yamabe flow](https://arxiv.org/pdf/1706.04917.pdf)
which said that
**Now for $k$ big enough $(k>n)$, the first eigenvalue of the oper... | https://mathoverflow.net/users/469129 | What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$ | A partial answer to your question. I would think it means:
$$
\sup\bigg(\| u(t,\cdot)\|\_{H^k\_0} : u(t,\cdot)=\exp(-t\Delta) u(0,\cdot),\textrm{ with } \| u(0,\cdot)\|\_{H^k\_0}=1\bigg).
$$
| 2 | https://mathoverflow.net/users/40120 | 425586 | 172,788 |
https://mathoverflow.net/questions/425560 | 2 | Let $[\omega]^\omega$ the collection of infinite subsets of $\omega$. We say that $E\subseteq [\omega]^\omega$ is *bipartite* if there is $d\subseteq \omega$ such that for all $e\in E$ the intersections $e\cap d$ and $e\cap (\omega\setminus d)$ are both non-empty. If $E\subseteq[\omega]^\omega$ is countable, then $(\om... | https://mathoverflow.net/users/8628 | Minimal cardinality of non-bipartite sub-family of $[\omega]^\omega$ | The cardinal $\mathfrak{nb}$ is equal to the [reaping number](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum#Splitting_number_%7F%27%22%60UNIQ--postMath-0000002D-QINU%60%22%27%7F_and_reaping_number_%7F%27%22%60UNIQ--postMath-0000002E-QINU%60%22%27%7F) $\mathfrak{r}$.
An *unsplit family* is a c... | 3 | https://mathoverflow.net/users/70618 | 425587 | 172,789 |
https://mathoverflow.net/questions/425526 | 9 | *Previously [asked at MSE](https://math.stackexchange.com/questions/4306473/is-there-a-finite-basis-for-the-equational-theory-of-the-orthocenter):*
Briefly speaking, I'm looking for a description of the equational theory of the **orthocenter function**, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) functio... | https://mathoverflow.net/users/8133 | Equational theory of the orthocenter | $\newcommand{\o}[0]{\mathsf{orth}}$No, these equations do not yield the complete theory of the orthocenter.
The identity
$$\o(\o(t,u,v),\o(t,u,w),u) = \o(\o(t,u,v),\o(t,v,w),v)$$
holds for the orthocenter (X(4)) but not for [X(74)](https://faculty.evansville.edu/ck6/encyclopedia/ETC.html) (the isogonal conjugate ... | 12 | https://mathoverflow.net/users/nan | 425590 | 172,791 |
https://mathoverflow.net/questions/303404 | 6 | The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, or for a connection obtained as the pullback of a connection on a vector bundle $E \to M$ *isomorphic to $TM$* via an is... | https://mathoverflow.net/users/74372 | Why torsion is only defined for linear connection on TM? | Actually, one can define torsion for Lie algebroids, and in particular vector bundles, eg. here: <https://arxiv.org/pdf/math/0105033.pdf>
It is defined as so: given a Lie algebroid $A\to M$ with anchor map $\alpha\,,$ choose a connection $\nabla\,.$ The torsion tensor is then defined as $T(X,Y)=\nabla\_{\alpha(X)}Y-\... | 1 | https://mathoverflow.net/users/40323 | 425593 | 172,794 |
https://mathoverflow.net/questions/425403 | 7 | Consider the alternating group graph, here defined as a Cayley graph on the alternating group $A\_n$ using the generating set $\{(1,2,3),(1,2,4),\dotsc,(1,2,n),(1,n,2),\dotsc,(1,4,2),(1,3,2)\}$. Note that when $n=3$, the graph reduces to a triangle.
Observing that the clique size is just $3$ (this can be seen by obse... | https://mathoverflow.net/users/100231 | 3-coloring the alternating group graph | So I believe that $\chi(A\_7) > 3$, but this relies on some computations that require verification.
But I will start by giving some SageMath code that computes the graph, finds an independent set of size $840$, and then shows that this independent set is not a colour class in any 3-colouring.
The code is a bit clun... | 5 | https://mathoverflow.net/users/1492 | 425616 | 172,796 |
https://mathoverflow.net/questions/425602 | 4 | Does there exist a function $f$ that satisfies all of the following three properties?
1. The function converts *an arbitrarily large* (empty, finite, countably/uncountably infinite) set of ordinals to a single ordinal;
2. If two sets $S$ and $T$ of ordinals are not equal, then two ordinals $f(S)$ and $f(T)$ are not e... | https://mathoverflow.net/users/122796 | Existence of a particular function that maps an arbitrary set of ordinals to a single ordinal |
>
> The existence of a function $f$ as specified in the question cannot be proved in ZFC. This follows from the following theorem and the well-known independence of $\mathrm{V = OD}$ (equivalently: $\mathrm{V = HOD}$) from $\mathrm{ZFC}$. Recall that $\mathrm{OD}$ is the class of ordinal-definable sets.
>
>
>
**... | 7 | https://mathoverflow.net/users/9269 | 425623 | 172,797 |
https://mathoverflow.net/questions/425485 | 2 | A Lie algebra is **complete** if its center is zero and all its derivations are inner. I would like to study a class of Lie algebras, in particular
>
> Let $C$ be the class of finite dimensional $2$-step solvable Lie algebras with trivial center (over $\mathbb{R}$ or $\mathbb{C}$). Is the completeness of the Lie al... | https://mathoverflow.net/users/56980 | Complete $2$-step solvable Lie algebras | $\DeclareMathOperator\g{\mathfrak{g}}\newcommand{\mk}{\mathfrak}$Here are some general facts.
**Proposition.** Let $\g$ be a metabelian (=2-step solvable) Lie algebra, finite-dimensional over a field of char. zero. If $\g$ has a trivial center then there exists an abelian subalgebra $\mk{a}$ of $\g$ such that $\g=\mk... | 2 | https://mathoverflow.net/users/14094 | 425636 | 172,800 |
https://mathoverflow.net/questions/425642 | 2 | **Setup/Motivation:**
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class of bi-Lipschitz smooth maps from $M$ to $N$. When is $\operatorname{bi-C}^{\infty}(M,N)$ dense in $\operatorname{C}^... | https://mathoverflow.net/users/469470 | Density of smooth bi-Lipschitz maps in smooth maps | Suppose that neither $M$ nor $N$ are compact and have dimensions both 2 or more.
Take an unbounded real valued function $f$ on $M$ and an unbounded geodesic $\gamma$ on $N$. Map $M$ to $N$ by taking each point $m \in M$ to the point $\gamma(t)$ where $t=f(m)$. This map is smooth but not approximable by any bi-Lipschitz... | 2 | https://mathoverflow.net/users/13268 | 425643 | 172,802 |
https://mathoverflow.net/questions/413531 | 4 | I believe the following is well known after talking to some experts, but I am unable to find a reference for the case with boundary.
Fix a field $F$ and an oriented $n$-manifold $(M,\partial M)$. We have a pairing $H\_\*(M) \otimes H\_{n-\*}(M,\partial M) \rightarrow F$ given by taking transverse representatives of t... | https://mathoverflow.net/users/134512 | Reference request for Poincaré–Lefschetz duality as an intersection pairing | This is proven for smooth manifolds by Goresky in "Whitney Stratified Chains and Cochains".
| 2 | https://mathoverflow.net/users/134512 | 425668 | 172,810 |
https://mathoverflow.net/questions/425672 | 4 | I stumbled on the following rather appealing trigonometric definite integral,
\begin{equation}
\int\_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)}
\end{equation}
for $a\in (-1,1)$ and $y\in[0,\pi]$. Does anyone know a reference or a simple proof?
| https://mathoverflow.net/users/47484 | Definite integral of power of sine ratio | Just after submitting the question I realised there is a rather simple Mellin transform solution to this problem, which I record below. I'd still be interested in a reference or generalizations of this identity.
Making the substitution $z = \sin(y-x)/\sin x = \sin y \cot x -\cos y$ we get the integral
\begin{align}... | 5 | https://mathoverflow.net/users/47484 | 425673 | 172,811 |
https://mathoverflow.net/questions/425669 | 4 | Fix $k$ a number field, and let $C$ be a smooth geometrically integral affine curve over $k$. We can "associate" to $C$ a semi-abelian variety in the following way:
One knows that $C$ is a finite number of (closed) points away from its smooth compactification, which is a projective curve $X$. Let these points be deno... | https://mathoverflow.net/users/172132 | Curves and semi-abelian varieties | Let me treat in some details the case $n=2$ — the general case is similar. Consider the nodal curve $Y$ obtained from $X$ by identifying $p\_1$ and $p\_2$. Then $J\_D$ is the Jacobian $JY$ of $Y$. Pulling back to $X$ gives an exact sequence
$$0\rightarrow \mathbb{G}\_m\rightarrow JY\rightarrow JX\rightarrow 0\,.$$
Such... | 9 | https://mathoverflow.net/users/40297 | 425674 | 172,812 |
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