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https://mathoverflow.net/questions/404208 | 1 | Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T\_1,T\_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T\_i)$ of $T\_i$ satisfies $\sigma(T\_i) \subset \mathbb{R}$, for $i=1,2$ (see Theorem $29.2$ in $[3]$). Suppose that $T\_1$ and $... | https://mathoverflow.net/users/156344 | A consequence of the Min-Max Principle for self-adjoint operators | I'm expanding my comment, in response to the OP's comment. Indeed, the case of just the lowest eigenvalue is perhaps not a good illustration of the full argument.
In general, let $u\_j$ be a normalized eigenvector for $\lambda\_j(T\_1)$, so $T\_1 u\_j=\lambda\_j(T\_1) u\_j$. Also make sure that the $u\_j$ are orthogo... | 0 | https://mathoverflow.net/users/48839 | 404254 | 165,802 |
https://mathoverflow.net/questions/404261 | 1 | Let $G=p^{n+m}.Q$ be an extension group of the special $p$-group $p^{n+m}$ by a group $Q$. Now $p^{n+m}=p^n{{}^\cdot}p^m$. How does one show that $\frac{G}{p^n}\cong p^m.Q$? Or equivalently that $G \cong p^n{{}^\cdot}(p^m{.}Q)$ (non-split extension of $p^n$ by $p^m{.}Q$?
| https://mathoverflow.net/users/148317 | On a quotient of a finite extension group $G=p^{n+m}.Q$ | I presume you mean that $G$ has a normal subgroup $P$ with $G/P \cong Q$, where $P$ is a special $p$-group with $Z(P) = [P,P] = \Phi(P)$ elementary abelian of order $p^n$, and $P/Z(P)$ elementary abelian of order $p^m$.
But $Z(P)$ is characteristic in $P$ and hence normal in $G$, so $G$ has the structure ${p^n}^. (p^... | 1 | https://mathoverflow.net/users/35840 | 404264 | 165,805 |
https://mathoverflow.net/questions/397791 | 7 | Given a function $f \in L^1 (\mathbb R)$, define the *roughness* $R\_f$ of $f$ at $x \in \mathbb R$ by
$$\DeclareMathOperator{\esssup}{\operatorname{esssup}}
R\_f (x) := \limsup\_{r \to 0+}\dfrac{r \esssup\_{y \in B\_r (x)} |f(y) - f(x)|}{\displaystyle\int\limits\_{B\_r (x)} |f(s) - f(x)| ds}
$$
where $\esssup$ deno... | https://mathoverflow.net/users/173490 | An equivalent condition for differentiability almost everywhere? | The answer is **negative**: If $f'(x) = 0$ "too often", then $R\_f$ may fail to be equal to one almost everywhere.
---
Let $C$ be a [fat Cantor set](https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set), let $I\_n = (a\_n, b\_n)$ ($n \geqslant 2$) be the sequence of all finite components of the... | 2 | https://mathoverflow.net/users/108637 | 404279 | 165,813 |
https://mathoverflow.net/questions/404218 | 2 | Let $S$ be a set with $\lvert S\rvert=\lvert\mathbb{R}\rvert$. Suppose it has subsets $S\_x$ indexed by $x\in \mathbb{R}$ with $\lvert S\_x\rvert=\lvert\mathbb{R}\lvert$ for each $x\in \mathbb{R}$. Suppose that
* for any $s\in S$ we have $\lvert\{x\in\mathbb{R}\mathrel\vert s\in S\_x\}\rvert=2$
* for any $x\neq y\in ... | https://mathoverflow.net/users/376005 | Include each point of continuum in a subset so that each subset gets finitely many points | Let $M : = \mathbb R^2 \setminus\{(x, y): x^2 + y^2 \leq 1\}$, $\Delta := \{(x, x) \in \mathbb R^2\}$, and let $h$ be any bijection from $\mathbb R$ to the circle $\{(x, y) \in \mathbb R^2: x^2 + y^2 = 1\}$.
Define $S := (\mathbb{R}^2 \setminus \Delta) \sqcup M \sqcup (\mathbb R/{\sim})$, where $x \sim y$ if and only... | 3 | https://mathoverflow.net/users/177751 | 404281 | 165,815 |
https://mathoverflow.net/questions/404245 | 8 | In the mod $p$ local Langlands correspondence for $\mathrm{GL}\_{2}(\mathbb{Q}\_{p})$, the irreducible supercuspidal representation $\left(\mathrm{ind}^{\mathrm{GL}\_{2}(\mathbb{Q}\_{p})}\_{\mathrm{GL}\_{2}(\mathbb{Z}\_{p})\mathbb{Q}\_{p}^{\times}}\mathrm{Sym}^{r}\overline{\mathbb{F}}\_{p}^{2}\right)/T$ of $\mathrm{GL}... | https://mathoverflow.net/users/34414 | A question about mod $p$ local Langlands for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ | You seem to be expecting that mod $p$ local Langlands should satisfy the same compatibilities as "conventional" local Langlands (for smooth representations of $GL\_2(\mathbf{Q}\_p)$ and $WD(\mathbf{Q}\_p)$ with coefficients in $\mathbf{C}$).
However, before you can even talk about reduction mod $p$, you need to check... | 6 | https://mathoverflow.net/users/2481 | 404291 | 165,817 |
https://mathoverflow.net/questions/404257 | 2 | I'm looking for an elegant way to show the following claim.
**Claim:** Let $m\_1, m\_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are $\sigma\_1 = \sqrt{\|m\_1\|\_2 + \left|\cos{\measuredangle \left( m\_1, m\_2 \right)}\right| \|m\_2\|\_2}$ and ... | https://mathoverflow.net/users/150065 | Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them | Since $\sigma\_1^2=\lambda\_+$ and $\sigma\_2^2=\lambda\_-$ are the two eigenvalues of the symmetric matrix product $MM^t$, we have $\lambda\_++\lambda\_-={\rm tr}\,MM^t=\|m\_1\|^2+\|m\_2\|^2$. Hence we may write WLOG
$$\lambda\_\pm=\tfrac{1}{2}\left(\|m\_1\|^2+\|m\_2\|^2\right)\pm\Delta.$$
To determine $\Delta$ we equ... | 2 | https://mathoverflow.net/users/11260 | 404301 | 165,822 |
https://mathoverflow.net/questions/404266 | 3 | The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1\_A$).
A similar story is true for $\mathbb{Z}$ and invertible elements, but let's instead tell it in the $\infty$-setting: namely, the $\in... | https://mathoverflow.net/users/130058 | Corepresentability of involutory objects in monoidal $\infty$-categories | $$Fun^{\otimes}(\mathbb Z/2, C) \simeq map\_{E\_1}(\mathbb Z/2, C^\simeq) \simeq map\_{E\_k}(\mathrm{Ind}\_{E\_1}^{E\_k}\mathbb Z/2, C^\simeq)$$
where $\mathrm{Ind}\_{E\_1}^{E\_k}$ denotes the left adjoint to the forgetful functor.
So $Inv$ is representable, and the natural $E\_{k-1}$-structure (see my comments for w... | 4 | https://mathoverflow.net/users/102343 | 404304 | 165,824 |
https://mathoverflow.net/questions/404310 | 2 | Let $\mathbb{F}\_{q^n}/\mathbb{F}\_q$ be an extension of finite fields.
Is a proper quotient of $\mathbb{F}\_{q^n}[x]$ considered as an $\mathbb{F}\_q$-algebra always a quotient of $\mathbb{F}\_q[x]$ (i.e. no extra generator is necessary)?
| https://mathoverflow.net/users/378181 | Is a proper quotient of $\mathbb{F}_{q^n}[x]$ considered as an $\mathbb{F}_q$-algebra always a quotient of $\mathbb{F}_q[x]$? | The answer is no. A counterexample: the quotient $\mathbb{F}\_4[x]/(x(x-1))$ is isomorphic to $\mathbb{F}\_4\times\mathbb{F}\_4$. If $\mathbb{F}\_2[x] / (f(x))$ were isomorphic to $\mathbb{F}\_4\times\mathbb{F}\_4$, $f(x)$ would need to be a product of two distinct irreducibles, each of degree two. But there is only on... | 5 | https://mathoverflow.net/users/5263 | 404317 | 165,830 |
https://mathoverflow.net/questions/404055 | 1 | Let $X$ is a strip between two different parallel lines $a$ and $b$ on a plane ($a,b\subset X$) and $h(x)=\min\limits\_{l\in \{a,b\}}\{d(x,l)\}$.
Let $(X,\*)$ be a topological group with the following property:
$$h(xy)\leq \max\{h(x),h(y) \}.$$
It is a locally compact, connected, simply connected, Hausdorff group. I th... | https://mathoverflow.net/users/175589 | Group structure on the strip | Such group does not exist. To derive a contradiction, assume that the strip $X=\mathbb R\times(-1,1)$ admits a continuous group operation $X\times X\to X$, $(x,y)\mapsto xy$, such that $h(xy)\le\max\{h(x),h(y)\}$ for all $x,y\in X$.
Let $c$ be any point on the central line $L=\mathbb R\times\{0\}$ and $f:X\to X$ be t... | 2 | https://mathoverflow.net/users/61536 | 404328 | 165,832 |
https://mathoverflow.net/questions/403960 | 4 | If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit vectors $e\_1,\dots,e\_n$? That is, under what conditions can I find $Q\succeq0$ such that $$(e\_i-e\_j)^TQ(e\_i-e\_j) = d(i... | https://mathoverflow.net/users/70190 | When does a finite metric induce a matrix norm? | Your quadratic form $Q$ is uniquely defined by $d$ on the hyperplane $H$ defined by $\sum x\_i=0$. Further, $Q|\_H\ge 0$ if and only if your metric space is isometric to a subset of a Eulcidean space.
| 2 | https://mathoverflow.net/users/1441 | 404331 | 165,833 |
https://mathoverflow.net/questions/404325 | 2 | In the [Young's lattice](https://en.wikipedia.org/wiki/Young%27s_lattice), the number of branches that connect the $N$'th layer to the $N+1$'th layer has the sequence:
$$
1,2, 4, 7, 12, 19, 30, 45, 67, 97, 139, \cdots
$$
Looking this up on OEIS, leads to [this result](https://oeis.org/A000070). This is nothing but the ... | https://mathoverflow.net/users/140290 | Number of branches between two layers of the Young's lattice | Contra what I originally thought, I’m not sure this is a general fact for all differential posets.
Nonetheless in the case of Young’s lattice it is easy to see this directly from the fact that if an element has $x$ edges coming in from below, it has $x+1$ going out above: we can always add a box in one more position ... | 3 | https://mathoverflow.net/users/25028 | 404332 | 165,834 |
https://mathoverflow.net/questions/404344 | 6 | What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?:
* Regard $G$ as a category with one object whose morphisms are the elements of $G$.
* Construct the simplicial sets $NG$ (i.e., the nerve of $G$) and ... | https://mathoverflow.net/users/137445 | Simplicial set construction of the classifying space | I believe that's called the Milgram bar construction:
* R.J. Milgram, *The bar construction and abelian $H$-spaces*, Illinois J. Math. 11 (1967), 242-250.
| 10 | https://mathoverflow.net/users/2926 | 404346 | 165,841 |
https://mathoverflow.net/questions/403296 | 1 | I encountered a sentence which says it is well known that problem
$$
\begin{cases}
-\Delta u =|u|^{p-1} u & in \,\, \Omega \\
u=0 & on \,\, \partial \Omega
\end{cases}
$$
has a solution for $1<p<\frac{N+2}{N-2}$ and doesn't have any solution for $p>\frac{N+2}{N-2}$.
The existence is okay by mountain pass theorem. But... | https://mathoverflow.net/users/76453 | Non-existence result for $p>\frac{N+2}{N-2}$ | It is not true that this equation always has no solutions in the supercritical case $p > \frac{N + 2}{N - 2}$.
The simplest counterexample is on an annulus, say $\Omega = B\_R \setminus B\_1$: in this case one may search for radial solutions by separating variables, which reduces to solving the second-order ODE
$$
-u... | 6 | https://mathoverflow.net/users/378654 | 404354 | 165,842 |
https://mathoverflow.net/questions/404348 | 0 | Suppose that $n$ is a natural number, $X$ is a set, and $S\subseteq X^{2}$ is a subset such that if $x,y\in X$, then there is a unique tuple $(x\_{0},\dots,x\_{n})$ where $x\_{0}=x,x\_{n}=y$ and $(x\_{i},x\_{i+1})\in S$ for $0\leq i<n$ (this condition is equivalent to saying that if $\chi\_{S}$ is the characteristic fu... | https://mathoverflow.net/users/22277 | Can this fixed point theorem generalize to infinite structures? | Here's one way to build an infinite counterexample with $n=2$ for simplicity:
We start with (say) $X\_0=\{0\}, S\_0=\emptyset$. Having defined $X\_m, S\_m$, we define $X\_{m+1}, S\_{m+1}$ as follows:
* To get $X\_{m+1}$, we add to $X\_m$ a fresh element $c\_{(x,y)}$ for each pair $(x,y)\in X\_m^2$ such that there i... | 2 | https://mathoverflow.net/users/8133 | 404355 | 165,843 |
https://mathoverflow.net/questions/404340 | 5 | The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the critical strip off the critical line?
This question came to my mind considering the sequence of trivial zeros in decre... | https://mathoverflow.net/users/13625 | Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line? | A result of Sarnak and Zaharescu, stated
in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong Siegel zero: [ProjectEuclid link](https://www.projecteuclid.org/journals/duke-mathematical-journal/volume-111/issue-3/Some-... | 19 | https://mathoverflow.net/users/766 | 404356 | 165,844 |
https://mathoverflow.net/questions/404350 | 1 | $\DeclareMathOperator\PL{PL}$Consider the group $\PL\_{n,n-1}$ of orientation preserving PL self-homeomorphisms of $\mathbb R^n$ that also preserve $\mathbb R^{n-1}$ pointwise. It is usually understood as a simplicial group whose $k$-simplices are PL self-homeomorphisms $\mathbb R^n\times\Delta^k\to \mathbb R^n\times \... | https://mathoverflow.net/users/9800 | Is $\operatorname{PL}_{n,n-1}$ contractible? | The answer is yes, see the discussion above with [Connor Malin](https://mathoverflow.net/users/134512/connor-malin).
| 1 | https://mathoverflow.net/users/9800 | 404359 | 165,845 |
https://mathoverflow.net/questions/404369 | 2 | Is there an uncountable integral domain such its every countable subset is contained in a finitely generated $\mathbb{Z}$-algebra?
| https://mathoverflow.net/users/378181 | Uncountable integral domain such that every countable subset is contained in a finitely generated $\mathbb{Z}$-algebra | 1- No. Indeed, let $A$ be an uncountable domain and $X$ a maximal algebraically independent subset (over the minimal subring $A\_0=\mathbf{Z}$ or $\mathbf{F}\_p$). Then $A$ is contained in an algebraic closure of the field of rational functions $\mathrm{Frac}(A\_0)(x:x\in X)$. In particular, it follows that $X$ is unco... | 5 | https://mathoverflow.net/users/14094 | 404373 | 165,850 |
https://mathoverflow.net/questions/402927 | 6 | **The situation:**
Let $X$ be a 2 dimensional normal quasi-projective $\mathcal{O}\_K$-scheme, where $K$ is an algebraic number field. Assume the following conditions on $X$:
1. $X$ is integral.
2. $X\_K$ is geometrically integral.
3. $X \to \textrm{spec}(\mathcal{O}\_K)$ is surjective.
Let $X\to \bar{X}$ an open i... | https://mathoverflow.net/users/352015 | Contraction of some surfaces over a ring of algebraic integers | The paper [2] is a seminar talk announcing the results of [3]. In this talk, I explain how to deduce the existence of integral points (theorem 1) from the contraction theorem (theorem 2). In turn, the contraction theorem is due to Artin [1] in the geometric case (surfaces over finite fields), but Raynaud explained me t... | 6 | https://mathoverflow.net/users/7666 | 404376 | 165,851 |
https://mathoverflow.net/questions/404296 | 4 | Let $X$ be the solution to the one dimensional SDE
$dX\_t = \mu(t, X\_t)dt + \sigma(t, X\_t) dW\_t$, for $t \in [0, T]$.
with $X\_0= x\_0$ a.s. for some $x\_0 \in \mathbb R$.
Here $W\_t$ denotes a standard Brownian motion, and we assume $\mu$ and $\sigma$ are Lipschitz continuous and uniformly bounded.
For ever... | https://mathoverflow.net/users/173490 | Conditioning an SDE on the event that the driving noise is small | The answer is yes, provided that you write your equation in Stratonovich form, rather than Itô form (and assuming that $\mu$ and $\sigma$ are sufficiently smooth in their arguments). The reason is that in one dimension the solution to the Stratonovich equation is a continuous map of $W$ in the sup-norm topology, [as ob... | 3 | https://mathoverflow.net/users/38566 | 404379 | 165,852 |
https://mathoverflow.net/questions/404357 | 2 | Let $X$ be a Banach space. If $X^{\*\*}$ is linearly isometric to $L\_{1}(\mu)$ for some $\sigma$-finite measue $\mu$, we shall say that $X$ is an $L\_{1}$-pre-bidual.
Question 1. What are the examples of $L\_{1}$-pre-bidual ?
Question 2. Are there any characterizations or even references about $L\_{1}$-pre-biduals... | https://mathoverflow.net/users/41619 | Banach spaces whose biduals are $L_{1}$ | If $X$ is an infinite dimensional Banach space such that $X^{\*\*}$ is isomorphic to $L^1(\mu)$ for some $\sigma$-finite measure, then $X^{\*\*}$ is non-reflexive, separable and has DPP (Dunford-Pettis property) since reflexivity, separability and DPP are isomorphic properties. This is not possible ([Banach spaces whos... | 3 | https://mathoverflow.net/users/164350 | 404386 | 165,853 |
https://mathoverflow.net/questions/404372 | 2 | We consider a sequence $u = (u\_k)\_{k\geq 1}$ such that $u\_k \geq 0$ for any $k \geq 1$. We assume that there exists a critical $p\_c \in \mathbb{R}$ such that, for any $q<p\_c <p$,
$$\sum\_{k=1}^\infty k^q u\_k < \infty \quad \text{and} \quad \sum\_{k=1}^\infty k^p u\_k = \infty.$$
I am interested in the asymptotic ... | https://mathoverflow.net/users/39261 | Asymptotic behavior of the moments of non-negative sequences | $\newcommand\ep\epsilon$The answer is no.
Indeed, let e.g.
$u\_k:=1$ if $k=k\_j:=2^{5^j}$ for some natural $j$, and $u\_k:=0$ otherwise. Then $p\_c=0$.
Take now any real $\ep>0$ and then take $p=2\ep$, so that condition $0< \epsilon <p-p\_c$ holds. Then for all large enough $j$ and $n=k\_{j+1}-1$ we have
$$n^{p-p\_... | 2 | https://mathoverflow.net/users/36721 | 404388 | 165,854 |
https://mathoverflow.net/questions/404391 | 3 | Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu\_g^{(f)}\in \mathrm{End}\_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$.
For monics $f\_1,f\_2\in A[x]$, how to directly prove that $\det \mu\_g^{(f\_1f\_2)}=\det\mu\_g^{(f\_1)}\det\mu\_g^{(f\_2)}$?
... | https://mathoverflow.net/users/69037 | Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices) | You have an exact sequence, $0\to A[x]/f\_1\stackrel{f\_2}{\to} A[x]/f\_1f\_2\to A[x]/f\_2\to 0$. This splits as $A$-modules and then multiplication by $g$ in the middle is just the diagonal matrix of multiplication by $g$ in the two factors. So, determinant multiplies.
| 4 | https://mathoverflow.net/users/9502 | 404394 | 165,855 |
https://mathoverflow.net/questions/404401 | 1 | Let $X\sim\mathcal{N}(\boldsymbol{\mu}\_1,\mathrm{\Sigma}\_1)$ and $Y\sim\mathcal{N}(\boldsymbol{\mu}\_2,\mathrm{\Sigma}\_2)$. Then it is know that $\mathbb{P}(X>\boldsymbol{t})\leq\mathbb{P}(Y>\boldsymbol{t})$ implies $\mu\_i\leq \mu^{\prime}\_i$ and $\sigma\_{ii} = \sigma^{\prime}\_{ii}$ (Theorem 10 of Muller 2001, A... | https://mathoverflow.net/users/120111 | Stochastic ordering of absolute multivariate normal random variables | $\newcommand\si\sigma$The condition $P(|X|>\boldsymbol{t})\le P(|Y|>\boldsymbol{t})$ implies $P(|X\_i|>t)\le P(|Y\_i|>t)$, for each $i$ and all real $t$; this follows by letting $t\_j=0$ for all $j\notin\{i\}$.
Fix any $i$ and, for brevity, let $m:=EY\_i$, $s:=\sqrt{Var\,Y\_i}$, $m\_1:=EX\_i$, and $s\_1:=\sqrt{Var\,X... | 1 | https://mathoverflow.net/users/36721 | 404404 | 165,857 |
https://mathoverflow.net/questions/404390 | 0 | In this post I present my variations of the problem involving Nagell-Ljunggren equation, that is explained in pages 10 and 11 of *Highlights in the Research Work of T. N. Shorey* by R. Tijdeman, from *Diophantine Equations*, (Editor) N. Saradha, Tata Institute of Fundamental Research, Narosa Publishing House (2008).
... | https://mathoverflow.net/users/142929 | Diophantine equations that involve Lehmer means with all digits equal to $1$ in their $x-$adic expansions | Conjecture 1 does not hold as for any even $n$, (2) has a solution:
$$(x,y,z)=(2,\frac23(2^n-1),2(2^n-1))$$
Likewise, Conjecture 2 fails as for $p=2$ and any even $n$, (1) has a solution:
$$(x,y,z)=(4,\frac2{15}(4^n-1),\frac25(4^n-1))$$
| 2 | https://mathoverflow.net/users/7076 | 404410 | 165,862 |
https://mathoverflow.net/questions/404324 | 3 | Let $A,B,C,D$ be the corners of a tetrahedron with positive volume and distinct sidelengths. Is there a positive $x$ and a planar straight-line embedding of a $K\_4$ graph with distinct vertices $A’,B’,C’,D’$ such that each edge of the tetrahedron is greater than the corresponding edge of the graph by exactly $x$?
| https://mathoverflow.net/users/31310 | Deflating a tetrahedron to a $K_4$ graph with equal changes to sidelengths | Yes, this is true. The main point is that the "first thing that goes wrong" cannot be two vertices coming together.
Let $a\_0$, $b\_0$, $c\_0$, $d\_0$, $e\_0$, $f\_0$ denote the edge lengths of the original tetrahedron, let $x$ be a variable, and let $a = a\_0 - x$, $b = b\_0 - x$, $c = c\_0 - x$, $d = d\_0 - x$, $e ... | 1 | https://mathoverflow.net/users/126667 | 404412 | 165,864 |
https://mathoverflow.net/questions/404362 | -1 | I am considering a *Principle of Ubiquity*, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:
For $\alpha(y,z)$ a first order condition so that $\forall y(\exists w(y\in w)\to \exists w (\{z|\alpha(y,z)\}\in w))$:
$\forall v(\exists t (v\in t)\to\exists... | https://mathoverflow.net/users/37385 | Ubiquity beyond infinity, transitive closure and the recursion theorem? | The answer seems to be affirmative.
Let an adapted version of ubiquity be as follows:
For $\alpha(y,z)$ a functional first order condition so that
$\forall y(\exists w(y\in w)\to \exists w(\{z|\alpha(y,z)\}\in w))$:
$\forall v(\exists w(v\in w)\to\exists w(\{u|\forall x(v\in x\wedge \forall n \forall y(((n,y)\i... | 0 | https://mathoverflow.net/users/37385 | 404416 | 165,866 |
https://mathoverflow.net/questions/404421 | 8 | By Dirchlet's hyperbola method, one can prove that the average number of divisors of integers $1 \leq n \leq X$ is $\log X$. This question concerns the number of integers $n \leq X$ such that the number of divisors, $d(n)$, is substantially larger than average. Indeed, what is known about the size of the set
$$\displ... | https://mathoverflow.net/users/10898 | Density of integers with many divisors | Theorem 1.11 and Theorem 1.22 of the paper by Norton, cited in the comment of Peter Humphries, show that for any fixed $A \ge \log 2$,
$$
\frac{X (\log\log X)^{O(1)}}{(\log X)^{B(A)}} \ll\_A
|\{1\le n\le X:d(n) \ge (\log X)^A\}| \ll\_A \frac{X}{(\log X)^{B(A)}},
$$
where
$$
B(A):=1+\frac{A}{\log 2}\left(\log\left(\fra... | 12 | https://mathoverflow.net/users/12947 | 404427 | 165,870 |
https://mathoverflow.net/questions/398543 | 0 | Let $M$ be a closed manifold and assume that is given a family of elliptic operators $L\_t,~t\in [0,1]$ and a smooth function $F :[a,b] \to \mathbb{R}$ such that for each $t$ the elliptic problem $L\_tu = F(u)$ has a classical solution $u : M \to \mathbb{R}$.
I would like to know which kind of methods and techniques ... | https://mathoverflow.net/users/94097 | Reference request and methods indication to the continuity of solutions to the problema $L_tu = F(u), ~t\in [0,1],$ and $L_t$ elliptic | This question is difficult to answer because the obstruction to continuous families of solutions is (usually) not technical but substantive, and has to do with the local uniqueness of solutions to the equation for a given $t$.
To see why, consider a simple example which illustrates the basic approach one might take: ... | 1 | https://mathoverflow.net/users/378654 | 404429 | 165,871 |
https://mathoverflow.net/questions/404435 | 4 | Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|\_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>0$?
Thanks,
| https://mathoverflow.net/users/50438 | Existence of a smooth compactly supported function | The answer is yes if $\epsilon<1$, and no when $\epsilon\geq 1$.
This follows from Carleman's quasianalyticity criterion, see for example, Hormander, Analysis of linear partial differential operators, Vol. I, Chap I, Section 1, Theorem 1.3.8.
(Carleman's original proof used Complex Analysis, and it was reproduced in ... | 15 | https://mathoverflow.net/users/25510 | 404438 | 165,873 |
https://mathoverflow.net/questions/404431 | 5 | We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align\*}
\mathsf{Forget} &\colon \mathsf{Ab} \to \mathsf{Sets},\\
\mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets}
\end{align\*}
given by $(A,\cdot\_A,1)\mapsto A$ and $A\mapsto\mathrm{Inv}(A)\overset{\ma... | https://mathoverflow.net/users/130058 | Ring spectra structures on a certain spectral analogue of $\mathbb{Z}/2$ | There are no left-unital multiplications on E. If there were, then for any element $x$ in $\pi\_n(E)$, we would have $x+x = 1 \cdot x + 1 \cdot x = (1+1) \cdot x = 0$ because all elements in $ \pi\_0 E$ are 2-torsion. This is not satisfied by the homotopy groups in your table.
| 8 | https://mathoverflow.net/users/360 | 404440 | 165,875 |
https://mathoverflow.net/questions/404449 | 1 | This is a (probably basic) question about the generator of a Markov process.
Let $(E,d)$ be a locally compact metric space. We consider a Feller process $X=(\{X\_t\}\_{t \ge 0},\{P\_x\}\_{x \in E})$ on $E$.
That is, for any $t>0$, the semigroup $P\_t$ of $X$ maps $C\_{\infty}(E)$ into itself. Here, $C\_\infty(E)$ den... | https://mathoverflow.net/users/68463 | On the generator of a Markov process | My second advice for those who work with generators of Markov processes is: *Use Dynkin's characteristic operator!* [\*]
---
If $f$ is in the domain of the generator $L$ of $X$, then
$$ L f(x) = \lim\_{B \to \{x\}} \frac{E\_x f(X(\tau\_B)) - f(x)}{E\_x \tau\_B} \, , $$
where the expression under the limit means, ... | 2 | https://mathoverflow.net/users/108637 | 404451 | 165,877 |
https://mathoverflow.net/questions/404230 | 5 | Let $\mathrm{cov}\_H(C\_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C\_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\max\{\mathrm{cov}(\mathcal M),\mathrm{cov}(\mathcal N)\}\le \mathrm{cov}(\mathcal E)\le\mathrm{cov}\_H(C\_2^\omega)\le... | https://mathoverflow.net/users/61536 | Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups? | Lyubomyr Zdomskyy proved that in the Laver model $\mathrm{cov}\_H(2^\omega)=\omega\_1<\mathfrak b=\mathfrak c$.
His argument used the following known *Laver property* of the Laver model $V'$: for every
function $f:\omega\to\omega$ in $V'$ upper bounded by some function $h:\omega\to\omega$ in the ground model $V$, the... | 3 | https://mathoverflow.net/users/61536 | 404455 | 165,879 |
https://mathoverflow.net/questions/404415 | 2 | I am looking to find a solution, or even just prove the existence of one, to the following system of linear PDEs. They come up in a construction I am trying to work out in symplectic geometry. Here $(r\_1, \theta\_1, r\_2, \theta\_2)$ are the polar coordinates on $\mathbb{R}^4$, and $\beta$ and $\gamma$ are smooth func... | https://mathoverflow.net/users/379942 | Existence of solution to a system of linear PDEs with boundary conditions | The general solution of your equations in a simply connected domain on which $r\_2\not=0$ and $r\_1\not=\pm1$ is
$$
\beta = \frac12 + \frac1{{(r\_1}^2{-}1)}\,
\left(\frac{\partial a}{\partial\theta\_1}+b(\theta\_1,r\_2)\right)
\quad\text{and}\quad
\gamma= \frac12 + \frac1{{r\_2}^2}\,
\left(\frac{\partial a}{\partial\th... | 4 | https://mathoverflow.net/users/13972 | 404456 | 165,880 |
https://mathoverflow.net/questions/404392 | 1 | Let $K$ be a perfect field, and let $f\_1, \ldots, f\_m \in K[X\_1,\ldots,X\_n]$ be polynomials. Consider the affine scheme
$$X = \mathrm{Spec}(K[X\_1,\ldots, X\_n]/(f\_1,\ldots,f\_m))$$
and let $N = \dim(X)$. Given a closed point $\mathfrak{m} \in X$, we define the *multiplicity* of $\mathfrak{m}$ to be $N!$ times the... | https://mathoverflow.net/users/125074 | Bound for multiplicities of closed points on scheme | Edited: the proof below assumes $k$ is algebraically closed. The proof for the multiplicity inequality has been added.
Given $x \in X := V(f\_1, \ldots, f\_m)$, let $k$ be the *local dimension* of $X$ at $x$ (i.e. $k$ is the maximum of the dimension of all irreducible components of $X$ containing $x$).
**Claim:** $... | 1 | https://mathoverflow.net/users/1508 | 404477 | 165,888 |
https://mathoverflow.net/questions/404472 | 4 | It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a complete set of relations that classify all 3-manifold in terms of links. In short, we have the map
$$\{\mbox{framed oriente... | https://mathoverflow.net/users/124549 | Normal form of framed links under Kirby moves | 1. It is a folklore result that geometrisation solves the homeomorphism problem for (compact, connected, oriented) three-manifolds. [Kuperberg](https://arxiv.org/abs/1508.06720) discusses this, and improves the running time to elementary recursive. So, use Snappy to convert the given Kirby diagrams to triangulations an... | 3 | https://mathoverflow.net/users/1650 | 404479 | 165,889 |
https://mathoverflow.net/questions/404478 | 3 | Let $G$ be a compact (Lie) group, and $H \subseteq H'$ two compact (Lie) subgroups. It is clear that we have an obvious surjective map of homogeneous spaces
$$
G/H \twoheadrightarrow G/H'.
$$
Will it be true in general that this fibration gives a fibre bundle? Will the fibre be isomorphic to the quotient space $H/H'$? ... | https://mathoverflow.net/users/326091 | Principal bundles from a fibration of homogeneous spaces | I call such bundles "homogeneous bundles", but it's not a totally standard terminology.
It is true that the map $G/H\rightarrow G/H'$ is a fiber bundle map with fiber $H/H'$. One way to see this is to start with the principal bundle $H\rightarrow G \rightarrow G/H$. The group $H$ naturally acts on $H/H'$, and so we h... | 4 | https://mathoverflow.net/users/1708 | 404483 | 165,890 |
https://mathoverflow.net/questions/404491 | 5 | Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define
$$ I\_j =\{ (\Lambda\_1,\Lambda\_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda\_1 \cap \Lambda\_2) \geq j \}. $$
These are an analytic subvarieties in $G(k,n) \times G(l,n)$. I would like to know if something is known about their hom... | https://mathoverflow.net/users/94647 | Intersection cycle in a product of Grassmannians | Let $V$ be the $n$-dimensional space such that $\Lambda\_i \subset V$. Then the condition $\dim(\Lambda\_1 \cap \Lambda\_2) \ge j$ is equivalent to the condition
$$
\mathrm{rank}(\Lambda\_1 \hookrightarrow V \to V/\Lambda\_2) \le k - j.
$$
This means that $I\_j$ is a degeneracy locus for the morphism
$$
\mathcal{U}\_1 ... | 8 | https://mathoverflow.net/users/4428 | 404493 | 165,896 |
https://mathoverflow.net/questions/404502 | 2 | Given a fixed $p \in \{3,4,5,\ldots\}$, we define the strictly increasing sequence $\{a\_k\}\_{k\in \mathbb N}$ as follows. We set $a\_{p,1}=1$ and for each $k>1$, we set $a\_{p,k}$ to be the least integer strictly greater than $a\_{p,k-1}$ such that
$$a\_{p,k}=b\_1+b\_2+\ldots+b\_p$$
for some $\{b\_j\}\_{j=1}^{p} \sub... | https://mathoverflow.net/users/50438 | On gaps in a sequence of integers | Unless I'm confused, this is true. The key fact is that your sequences are just infinite arithmetic progressions, i.e. $a\_{p,k} = 1 + (p-1)(k-1)$ for all $p,k$.
We can prove this formula by strong induction on $k$: clearly $a\_{p,1} = 1$ by definition. Assume that $a\_{p,k} = 1 + (p-1)(k-1)$ for $k < K$. Then all $a... | 4 | https://mathoverflow.net/users/116357 | 404510 | 165,901 |
https://mathoverflow.net/questions/404459 | 4 | Let $G$ be a simple (i.e. *every proper normal subgroup is discrete*) simply connected compact Lie group. Define **the degree of $k$-nilpotence** of $G$ to be the Haar measure of the set
$$\{(x\_1,\dotsc,x\_{k+1}): [x\_1,\dotsc,x\_{k+1}]=1\}.$$
($[x,y]=x^{-1}y^{-1}xy$ and $[x\_1,\dotsc,x\_{k+1}]:=[[x\_1,\dotsc,x\_k],x\... | https://mathoverflow.net/users/84700 | Can the degree of $k$-nilpotence of a simple simply connected compact Lie group be in $(0,1)$? | Here's a positive answer of the question for arbitrary compact Lie groups. For a group $G$, denote $W\_k(G)=\{x\in G^k:[x\_1,\dots,x\_k]=1\}$.
>
> Let $G$ be a compact Lie group. Then $W\_k(G)$ has nonzero Haar measure for some $k\ge 1$ if and only if $G^0$ is a torus.
>
>
>
One direction is obvious: if $G^0$ ... | 5 | https://mathoverflow.net/users/14094 | 404514 | 165,903 |
https://mathoverflow.net/questions/404518 | 1 | Consider infinite matrices of the form
$$\left(
\begin{array}{ccccc}
a\_0 & a\_1 & a\_2 & a\_3 & . \\
0 & a\_0 & a\_1 & a\_2 & . \\
0 & 0 & a\_0 & a\_1 & . \\
0 & 0 & 0 & a\_0 & . \\
. & . & . & . & . \\
\end{array}
\right)$$
The elements on each diagonal coincide.
My questions are:
* Do they form a commut... | https://mathoverflow.net/users/10059 | What are the properties of this set of infinite matrices and operations on them? | If the matrices have entries from a (unital) ring $R$ then the set of such matrices is isomorphic to $R[[x]]$, the ring of formal power series over $R$. To see this, observe that the map sending the infinite matrix with $a\_0 = 0$, $a\_1 = 1$ and $a\_k = 0$ for $k \ge 2$ to $x$ is a ring isomorphism.
This also answer... | 5 | https://mathoverflow.net/users/7709 | 404519 | 165,904 |
https://mathoverflow.net/questions/404482 | 2 | Let $\mathrm{sSet}^+ = \mathrm{sSet}^+\_{/ \Delta^0}$ be the model category of marked simplicial sets over the point. By Theorem 3.1.5.1 in Higher Topos Theory, this model category is Quillen equivalent to $\mathrm{sSet}$ with Joyal's model structure. The fibrant objects of $\mathrm{sSet}^+$ are the quasicategories in ... | https://mathoverflow.net/users/84063 | Fibrations of fibrant marked simplicial sets | Yes, this is true. There are various ways to prove this. Here's the shortest argument I can think of. One direction is easy to prove, so let's prove the other direction.
Let $U \colon \mathbf{sSet}^+ \to \mathbf{sSet}$ denote the functor that forgets markings. We will use that the restriction of this functor to the f... | 3 | https://mathoverflow.net/users/57405 | 404540 | 165,909 |
https://mathoverflow.net/questions/254887 | 7 | *In retrospect the original version of this question was impossibly bloated. Here's a better version:*
There are many results about when first-order sentences are preserved by algebraic operations on model classes; for example, in first-order logic the sentences preserved by taking substructures are those semanticall... | https://mathoverflow.net/users/8133 | Preservation results in abstract logics | EDIT: Now understanding "semantically equivalent", I change my answer (my previous answer, to a different question, is further below)...I'm working in ZFC.
Yes, there is such a computable set $X$; that is, $X$ is a set of productive second order sentences, and for every productive second order $\mathcal{L}$-sentence ... | 2 | https://mathoverflow.net/users/160347 | 404543 | 165,910 |
https://mathoverflow.net/questions/404552 | 0 | Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via [Fourier series](https://en.wikipedia.org/wiki/Fourier_series) as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \longrightarrow \mathbb{C}$ is a periodic function and is written as
$$
f(x)=\sum\_{n \in ... | https://mathoverflow.net/users/156344 | Inequality involving the fractional Laplacian | If true, this would follow from the integral expression for the fractional Laplacian:
$$(-\Delta)^{s/2} f(x) = \int\_{-\pi}^\pi (f(x) - f(y)) \nu(x - y) dy$$
for an appropriate kernel $\nu$. But, unfortunately, the claimed inequality is false: if, for example, $f$ is a non-zero odd function, then
$$(-\Delta)^{s/2} f(0)... | 3 | https://mathoverflow.net/users/108637 | 404557 | 165,915 |
https://mathoverflow.net/questions/404547 | 0 | Let $E$ be a separable $\mathbb R$-Banach space and $\lambda\_i$ be a finite symmetric measure on $\mathcal B(E)$ with $\lambda\_i(\{0\})=0$ and $$\int\_B1-\cos\langle x,x'\rangle\:\underbrace{(\lambda\_1-\lambda\_2)}\_{=:\:\sigma}({\rm d}x)=0\tag1$$ for all $B\in\mathcal B(E)$ and $x'\in E'$.
>
> How can we conclu... | https://mathoverflow.net/users/91890 | If $\lambda_i$ is symmetric with $\lambda_i\{0\}=0$, why does $\int_B1-\cos\langle x,x'\rangle\:(λ_1-λ_2)({\rm d}x)=0$ imply $λ_1=λ_2$? | $\newcommand\B{\mathcal B}$Let $C:=E\setminus\{0\}$, so that $g>0$ on $C$. Let $\rho(B):=\sigma(B)$ for all $B\in\B(C)$, so that $\rho$ is a signed measure defined on $\B(C)$ such that
$$\int\_B g\,d\rho=0 \tag{1}$$
for all $B\in\B(C)$.
By the [Hahn decomposition theorem](https://en.wikipedia.org/wiki/Hahn_decomposit... | 2 | https://mathoverflow.net/users/36721 | 404563 | 165,916 |
https://mathoverflow.net/questions/404529 | 4 | I am wondering how to prove the below Fourier transform is non-negative? I did much simulation and it seems to be non-negative.
$$\int\_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}<p<1$$
| https://mathoverflow.net/users/381188 | Fourier-positivity of a certain function | In the OP, it assumed that $1/2<p<1$. Let us first show that, actually, the desired conclusion holds for $p\in(0,1/2]$. Let
$$h(a):=\int\_0^\infty \frac{e^{-at^p}}a\,\cos(tx)\,dt.$$
As noted by Johannes Hahn, it suffices to show that $h$ is decreasing.
We have
$$-h'(a)=\int\_0^\infty g(t)\cos(tx)dt,$$
where
$$g(t)... | 3 | https://mathoverflow.net/users/36721 | 404565 | 165,918 |
https://mathoverflow.net/questions/404523 | 4 | Let $\Gamma$ be a torsion group (i.e. every element has finite order). I am interested in understanding central extensions of the form:
$\require{AMScd}$
\begin{CD}
0 @>>> \mathbb{R}^n @>\exp>> G @>\pi>> \Gamma @>>> 1\\
\end{CD}
Equivalently, I want examples of groups $\Gamma$ with non-trivial classes in $H^2(\Gamma,\m... | https://mathoverflow.net/users/381262 | Central extensions of torsion groups by $\mathbb{R}^n$ | The paper S. I. Adyan and V. S. Atabekyan, V. S.
*Central extensions of free periodic groups*,
Mat. Sb. 209 (2018), no. 12, 3–16; translation in
Sb. Math. 209 (2018), no. 12, 1677–1689
proves that if $n\geq 665$ is odd and $m\geq 2$, then the Schur multiplier $H^2(B(m,n),\mathbb Z)$ for the free Burnside group $B(m,n)$... | 3 | https://mathoverflow.net/users/15934 | 404591 | 165,925 |
https://mathoverflow.net/questions/400632 | 12 | *(Below I'm thinking only about computably axiomatizable set theories extending $\mathsf{ZFC}$ which are arithmetically, or at least $\Sigma^0\_1$-, sound.)*
Say that a theory $T$ is omniscient iff $T$ **proves that** the following holds:
>
> For every formula $\varphi(x,y)$ there is some formula $\psi(z,y)$ such... | https://mathoverflow.net/users/8133 | Is this definability principle consistent? | There is a consistent omniscient theory, at least assuming the consistency of a Woodin limit of Woodin cardinals.
The Maximality Principle (MP) asserts that if a sentence is forceable in $V$, it is forceable in every generic extension of $V$. In other words, if a sentence can be forced to be indestructible by set for... | 8 | https://mathoverflow.net/users/102684 | 404594 | 165,927 |
https://mathoverflow.net/questions/404532 | 4 | Suppose that $f:\Bbb R^2\to\Bbb R$ is a *continuous non-linearity* and consider the following semi-linear elliptic PDE given by:
$$-\Delta u=f(x,u),\;\;x\in\Omega\subset\Bbb R^n,\tag{1}\label{1}$$
To avoid the mention of critical Sobolev exponents and to narrow down the scope of the answer, let us assume that $n=2$ and... | https://mathoverflow.net/users/105925 | Regularity of weak solutions to semi-linear elliptic PDEs | There is a "standard" bootstrap argument which can be used to show regularity for semilinear equations. I sketch it here under the assumption that $|f(x, u)| \leq C(1 + |u|^p)$ for some $0 < p < \frac{2n}{n-2}$ in $n \geq 2$ (I know the question was posed in $n = 2$ but it is useful to see the role played by the critic... | 4 | https://mathoverflow.net/users/378654 | 404596 | 165,929 |
https://mathoverflow.net/questions/404037 | 2 | In a non weighted graph, the adjacency matrix ($A$) raised to the power $k$ will return the number of k-step paths between nodes $i$ and $j$ at the entry $a\_{ij}$. Is there an equivalent for weighted graphs? I.e. to obtain the *sum of the cumulative weights of all paths* between node pairs?
Applying the same approac... | https://mathoverflow.net/users/373251 | Solution to the sum of k-step path lengths between node pairs in directed weighted graphs | Let $A^k\_{i,j} = (n^k\_{i,j}, s^k\_{i,j})$ where $n^k\_{i,j}$ is the number of paths of length $k$ between $i$ and $j$, and $s^k\_{i,j}$ is the sum of the lengths between these paths.
$A^1\_{i,j} = (1, w\_{i,j})$ if $ij$ is an edge, else $A^1\_{i,j}=(0,0)$
The idea of the recurrence is that a $(k+k')$-step path ... | 1 | https://mathoverflow.net/users/381833 | 404611 | 165,932 |
https://mathoverflow.net/questions/404573 | 2 | Let $\mathcal{G} = G\_1 \rightrightarrows G\_0$ be a Lie Groupoid (although I am also interested in groupoids internal to other sites), the stack associated to $\mathcal{G}$, which is sometimes denoted $B \mathcal{G}$ is defined as the projection from the category of principal $\mathcal{G}$-bundles to $\mathsf{Man}$. H... | https://mathoverflow.net/users/124010 | Necessary and sufficient conditions for a Lie groupoid to present a stack | Note that $\hat r\mathcal{G}$ is always a prestack. This is basically equivalent to the fact that $C^\infty(-,G\_1)$ is a sheaf, and it means that your functor
$$
\hat r\mathcal{G}(U) \to \mathrm{holim}(...)
$$
can only fail to be essentially surjective.
Essential surjectivity of this functor is equivalent to saying... | 2 | https://mathoverflow.net/users/3473 | 404613 | 165,934 |
https://mathoverflow.net/questions/404616 | 6 | This problem is derived from [this post](https://mathoverflow.net/questions/403797/a-variation-of-the-ryll-nardzewski-fixed-point-theorem).
Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are equivalent.
(i) Every $... | https://mathoverflow.net/users/7591 | Trans-amenability of group actions | This is precisely what was called **$L$-amenability** by Kaimanovich and **lamenability** by Bartholdi (who were inspired by [Infinitely supported Liouville measures of Schreier graphs](https://mathscinet.ams.org/mathscinet-getitem?mr=3844999) of Juschenko and Zheng; "L" here stands for Liouville). In a sense, the afor... | 12 | https://mathoverflow.net/users/8588 | 404624 | 165,936 |
https://mathoverflow.net/questions/404384 | 0 | For any set $X$, we let $[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$.
If $G=(V,E)$ is a simple, undirected graph, and $v\in V$, let $N(v) = \{z\in V: \{v,z\}\in E\}$. Given any $v\in V$, we use the following notation:
1. Let $G\setminus\{v\} := (V\setminus \{v\}, E \cap [V\setminus\{v\}]^2)$, and
2. if $w\in V$ with... | https://mathoverflow.net/users/8628 | Two kinds of vertex-criticality | Unless I'm mistaken, your definition of $(G\setminus \{v\})^w$ is symmetrical, and generally referred as an edge-contraction, so collapse critical is the same as edge-contraction critical.
[this graph](https://math.stackexchange.com/questions/2430688/vertex-critical-graph-with-at-least-one-non-critical-edge) is verte... | 1 | https://mathoverflow.net/users/381833 | 404627 | 165,937 |
https://mathoverflow.net/questions/404587 | 1 | Let $X\_0/\mathbb F\_q$ be a variety, and let $\mathcal F$ be a Weil sheaf on $X := (X\_0)\_{\overline{\mathbb{F}\_q}}$ that is pure of weight $n$. If $j < n$, does the weight $j$ piece of $H^i\_c(X,\mathcal F)$ necessarily vanish for all $i$?
I thought this was true, but I have made some computations that seem to co... | https://mathoverflow.net/users/382874 | Lowest weight of compactly supported cohomology with coefficients | This is not true.
If you take an open curve $U \subset C$ and a pure sheaf $\mathcal F$ on $U$ of weight $w$, with $j$ the open immersion $U \to C$ and $i$ the complementary closed immersion, then the exact sequence $$ 0 \to j\_! \mathcal F \to j\_\* \mathcal F \to i\_\* i^\* j\_\* \mathcal F \to 0$$ induces a long e... | 3 | https://mathoverflow.net/users/18060 | 404631 | 165,938 |
https://mathoverflow.net/questions/404546 | 40 | As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. [this question](https://mathoverflow.net/questions/114943/where-are-the-second-and-third-generation-proofs-of-the-classification-of-fin... | https://mathoverflow.net/users/120914 | Known and fixed gaps in the proof of the CFSG | Here is an answer from my point of view, immersed as I am -- Geoff
is right -- in the second generation project. First, a few general
comments. Our overriding purpose has been to expound a coherent proof
of CFSG that is supported completely by what we call ``Background
Results,'' an explicit and restricted list of publ... | 39 | https://mathoverflow.net/users/99221 | 404644 | 165,941 |
https://mathoverflow.net/questions/404520 | 5 | I am interested in proving an upper bound (expressed as a power of $N$, with $N\rightarrow\infty$ ) for the number of elements of the set
$$
A\_N=\{(k,l,m,n)\in([N,2N]\cap\mathbb{Z})^4: |k^2+l^2-m^2-n^2|\le|k+l-m-n|\}.
$$
My intuition is that the inequality defining this set cannot hold too often unless the quadruple e... | https://mathoverflow.net/users/157356 | A bound for the number of integer solutions to a simple inequality | $|A\_N|$ has order of magnitude $N^3$:
We use the change of variables $x=k-m$, $y=k+m$, $u=n-l$, $v=n+l$, so that the inequality becomes $|xy-uv|\le |x-u|$. If $x=0$ or $u=0$, there are clearly $\ll N^3$ solutions.
Given $x\ge u \ge 1$ and $y$, and assuming $xy\ge uv$, the
variable $v$ must satisfy
$$
v \in [2N+u,4N-u]... | 9 | https://mathoverflow.net/users/12947 | 404645 | 165,942 |
https://mathoverflow.net/questions/404625 | 12 | A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi\_\*(O\_{\tilde X})$
vanish for all $i>0$. Suppose that $X$
has isolated rational singularity,
and $\tilde X\rightarrow X$ its resolution.
I expect that the fiber ... | https://mathoverflow.net/users/3377 | Does a resolution of a rational singularity have rationally connected fibers? | No. For instance the cone over an Enriques surface (with respect to any projective embedding) has rational singularity, but Enriques surface is not rationally connected.
| 15 | https://mathoverflow.net/users/4428 | 404646 | 165,943 |
https://mathoverflow.net/questions/404647 | 6 | $\DeclareMathOperator\cd{cd}$Recall that the **rational cohomological dimension** of a group $G$ is the supremum of the set of integers $k$ such that there exists a $\mathbb{Q}[G]$-module $M$ with $H^k(G;M) \neq 0$. Denote this by $\cd\_{\mathbb{Q}}(G)$.
If $G$ is a finite group, then it is easy to see that $\cd\_{\m... | https://mathoverflow.net/users/384401 | Rational cohomological dimension of a locally finite group | **Edited version.** If $G$ is a countable infinite locally finite group, then the rational cohomological dimension is exactly $1$. The rational cohomological dimension for an infinite group is never $0$ because if $\mathbb Q$ is a projective $\mathbb QG$-module, then you need an idempotent $e$ with $\mathbb QGe\cong \m... | 4 | https://mathoverflow.net/users/15934 | 404649 | 165,945 |
https://mathoverflow.net/questions/404607 | 10 | A **state sum model** is a smooth invariant defined on smooth triangulated, or PL manifolds, by summing a local partition function over labels attached to the elements of the triangulation.
Typical examples in 4d are the [Crane-Yetter invariant](https://ncatlab.org/nlab/show/Crane-Yetter+model) on ribbon fusion categ... | https://mathoverflow.net/users/13767 | Are there 4d state sum models, extended TQFTs or chain mail invariant that detect smooth structures? | [This MO answer](https://mathoverflow.net/a/362517/184) by Arun Debray gives an example in the unoriented case where two specific homeomorphic manifolds can be distinguished by a specific TFT of this kind.
In general all these constructions produce "semisimple" TFTs and it has been shown by David Reutter that in the ... | 7 | https://mathoverflow.net/users/184 | 404652 | 165,946 |
https://mathoverflow.net/questions/404640 | 6 | Given a probability $p \in (0,1)$ and parameter $\alpha \in (0,1)$, is there an entropy-based proof which yields a good upper bound for the sum
$$\sum\_{\ell = 0}^{\alpha n} \binom{n}{\ell}p^\ell(1-p)^{n-\ell}$$
when $n$ is large?
When $p = 1/2$, there is very simple proof (for example, see section 3.1 of [this paper... | https://mathoverflow.net/users/138628 | Is there an entropy proof for bounding a weighted sum of binomial coefficients? | Yes, if $\alpha<p$ (if $\alpha>p$, the sum is almost 1). To see this, write
$$
\sum\_{\ell = 0}^{\alpha n} \binom{n}{\ell}p^\ell(1-p)^{n-\ell}\leqslant t^{-\alpha n}(pt+(1-p))^n
$$
for every $t\in (0,1]$. Choose a positive $t=t\_0$ for which RHS is minimal possible, taking the logarithmic derivative equal to 0 we get $... | 7 | https://mathoverflow.net/users/4312 | 404654 | 165,947 |
https://mathoverflow.net/questions/404570 | 8 | **Setup/Notation:**
Let $n,m\in \mathbb{N}$ and let $C(\mathbb{R}^n,\mathbb{R}^m)$ be the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ equipped with the compact-open topology. Let $\mathcal{I}(\mathbb{R}^n,\mathbb{R}^m)$ be the subset of $C(\mathbb{R}^n,\mathbb{R}^m)$ consisting of injective func... | https://mathoverflow.net/users/36886 | The closure of the set of injective continuous functions | Kuratowski proved in [Sur les théorèmes du „plongement" dans la théorie de la dimension. *Fundamenta Mathematicae* 28.1 (1937): 336-342](http://matwbn.icm.edu.pl/ksiazki/fm/fm28/fm28137.pdf) that the set of embeddings of an at most $n$-dimensional separable metrizable space into $\mathbb{R}^{2n+1}$ contains a dense $G\... | 5 | https://mathoverflow.net/users/5903 | 404659 | 165,949 |
https://mathoverflow.net/questions/404656 | 2 | A few procrastinal computations motivated by [Four infinite series involving Riemann zeta function](https://mathoverflow.net/questions/401468/four-infinite-series-involving-riemann-zeta-function) suggest the identity
$$\tan\left(\frac{\kappa-1}{\kappa+1}\frac{\pi}{2}\right)=\frac{1}{\pi}\sum\_{n=1}^\infty \frac{\kappa^... | https://mathoverflow.net/users/4556 | A expression for the tangent function involving $\zeta(n),n=2,3,\ldots$ | When I was writing my answer, Dan Romik answer appeared. Mine is the same but with more detail.
We have
$$\log\Gamma(1-x)=\gamma x+\sum\_{n=2}^\infty\zeta(n)\frac{x^n}{n}$$
(this is known and is also an exercise in complex analysis).
Hence by differentiation
$$-\frac{\Gamma'(1-x)}{\Gamma(1-x)}-\gamma=\sum\_{n=2}^\inf... | 2 | https://mathoverflow.net/users/7402 | 404660 | 165,950 |
https://mathoverflow.net/questions/404629 | 2 | Let $r$ be the rank function of a matroid. If the matroid is representable (over a field), then $r$ must satisfy Ingleton's inequalities. On the other hand, there are matroids that satisfy Ingleton's inequalities but are still not representable (already on $8$ elements such examples exist). However, not satisfying Ingl... | https://mathoverflow.net/users/36563 | Non-representable matroids and Ingleton's inequality | I don't know the smallest such example, but here is an example with 14 elements.
* Let F be the Fano matroid: the 7-element matroid represented over any field of characteristic 2 by the set of non-zero vectors in $\{0,1\}^3$.
* Let N be the non-Fano matroid: the 7-element matroid represented over any field of charact... | 2 | https://mathoverflow.net/users/8049 | 404663 | 165,952 |
https://mathoverflow.net/questions/404530 | 12 | Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd.
```
Example:
a(3) = 3: 123, 12|3, 3|12.
a(4) = 5: 1234, 124|3, 3|124, 12|34, 34|12.
```
See [A290384](http://oeis.org/A290384) for details.
Motivated by [Question 403336](https://mathoverflow.net/question... | https://mathoverflow.net/users/287674 | Set partitions and permanents | Let me outline an approach for computing permanents in these conjectures. For the sake of concreteness, I will prove Conjecture 1 for an odd $n$. The matrix here is the sum of the following two 0-1 matrices (using [Iverson bracket](https://en.wikipedia.org/wiki/Iverson_bracket) notation):
$$A:=\big([2j-k \geq 1]\big)\_... | 8 | https://mathoverflow.net/users/7076 | 404669 | 165,953 |
https://mathoverflow.net/questions/403336 | 3 | Inspired by [Question 402572](https://mathoverflow.net/questions/402572), I consider the permanent of matrices
$$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]\_{1\le j,k\le n},$$
where $n$ is a positive integer and $\operatorname{sgn}$ is the sign-function.
W... | https://mathoverflow.net/users/287674 | Tangent numbers, secant numbers and permanent of matrices | The conjecture has been proved! See the preprint *Proof of five conjectures relating permanents to combinatorial sequences* by Fu, Lin and me available from <http://arXiv.org/abs/2109.11506>.
| 4 | https://mathoverflow.net/users/124654 | 404675 | 165,955 |
https://mathoverflow.net/questions/402572 | 21 | Motivated by [Question 402249](https://mathoverflow.net/questions/402249) of [Zhi-Wei Sun](https://mathoverflow.net/users/124654/zhi-wei-sun), I consider the permanent of matrices
$$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]\_{1\le j,k\le n-1},$$
where $n$ is an odd integer grea... | https://mathoverflow.net/users/287674 | Euler numbers and permanent of matrices | The conjecture and the added Conjectures 1 and 2 have been proved.
See the preprint *Proof of five conjectures relating permanents to combinatorial sequences* by Fu, Lin and me available from <http://arXiv.org/abs/2109.11506>.
| 5 | https://mathoverflow.net/users/124654 | 404676 | 165,956 |
https://mathoverflow.net/questions/404683 | 4 | Can you provide a proof for the following claim:
>
> $$-\displaystyle\sum\_{n=1}^{\infty}\frac{J\_k(n)}{n} \cdot \ln\left(1-x^n\right)=\frac{x \cdot A\_{k-1}(x)}{(1-x)^k} \quad \text{for} \quad |x| < 1 \quad \text{and} \quad k>1$$
> where $J\_k(n)$ is the [Jordan's totient function](https://en.wikipedia.org/wiki/Jo... | https://mathoverflow.net/users/88804 | An infinite series involving Jordan's totient function | The right hand side evaluates to
$$\sum\_{m=0}^{\infty} m^{k-1} x^m,$$
and so it remains to verify that the coefficient of $x^m$ in the l.h.s. is also $m^{k-1}$.
By differentiating the l.h.s., we have
\begin{split}
[x^m]\ \bigg(-\sum\_{n\geq 1} \frac{J\_k(n)}{n} \ln(1-x^n)\bigg) &= \frac1m [x^{m-1}]\ \sum\_{n\geq 1} ... | 6 | https://mathoverflow.net/users/7076 | 404685 | 165,960 |
https://mathoverflow.net/questions/404619 | 4 | Let $X$ be a compact metric space (without isolated points). The $\infty$-Wasserstein distance $W\_\infty$ on the space of Borel probability measures on $X$ can be described as $$W\_\infty(\mu,\nu) = \inf\{r>0 \mid \mu(U)\le\nu(U\_r)\,\forall \text{ open } U\subseteq X\},$$ where $U\_r=\{x\in X \mid d(x,U)\le r\}$. The... | https://mathoverflow.net/users/384059 | The infinity Wasserstein distance $W_\infty$ and the weak topology | First, I interpret your condition that "$X$ has no isolated points" in the following ways: First, every ball $B(x, \varepsilon)$ has non-empty interior. This means, in particular, that we can find arbitrarily fine partitions of $X$ with sets that each have non-empty interior. And second, if we have $X = \Omega\_1 \stac... | 3 | https://mathoverflow.net/users/106046 | 404689 | 165,961 |
https://mathoverflow.net/questions/404688 | 7 | Let $(X\_i)\_{i\in \mathbb{N}}$ iid random variables such that there exists $\alpha>0$ where $\mathbb{P}\left(X\_1\in [x,x+1]\right)\leq \alpha$ for all $x\in \mathbb{R}$. Assume $\alpha$ small enough, does there exist a universal constant $C>0$ so that $$\mathbb{P}\left(\sum\_{i=1}^N X\_i\in [x,x+1]\right)\leq \frac{C... | https://mathoverflow.net/users/99045 | Regularity for the sum of iid random variables | Yes, this is a special case of an inequality by Kesten (see [Theorem 2 and Corollary 1](https://www.mscand.dk/article/view/10950/8971)).
In particular, letting $L=\lambda=:t$ in that Corollary 1, we get the following:
>
> Let $X\_1,\dots,X\_n$ be independent identically distributed random variables, with $S\_n:=\... | 7 | https://mathoverflow.net/users/36721 | 404697 | 165,964 |
https://mathoverflow.net/questions/404699 | 1 | I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what I am not seeing.
Let me sketch the situation. Given a cooperad $(C,\Delta,\epsilon)$ and a operad $(P,\gamma,\eta)$, ... | https://mathoverflow.net/users/123496 | Differential of the Twisted complex for algebraic operads | Since $\alpha$ is of degree $-1$, these terms come in pairs appearing with opposite signs that cancel each other. In other words, the (co)associativity for (co)operads has a sequential axiom and a parallel axiom, and these terms vanish because of the parallel axiom. A nice way to see that is to draw these as trees and ... | 3 | https://mathoverflow.net/users/1306 | 404701 | 165,965 |
https://mathoverflow.net/questions/404664 | 9 | Let $\nu\_p(n)$ denote the $p$-adic valuation of $n$, i.e. the highest power of $p$ dividing $n$.
Consider the following two $q$-series formed by infinite products
$$\prod\_{n\geq1}\left(\frac{1+q^n}{1-q^n}\right)^2=\sum\_{k\geq0}a\_k\,q^k \qquad \text{and} \qquad
\prod\_{n\geq1}\left(\frac{1+q^n}{1-q^n}\right)^n=\su... | https://mathoverflow.net/users/66131 | $2$-adic valuations: a tale of two $q$-series | First notice that
$$\frac{1+q^n}{1-q^n} = 1 + 2\frac{q^n}{1-q^n}.$$
Computing modulo $8$, we have
$$\left(\frac{1+q^n}{1-q^n}\right)^2 \equiv 1 + 4\frac{q^n}{1-q^{2n}}\pmod8.$$
Correspondingly,
$$\prod\_{n\geq 1}\left(\frac{1+q^n}{1-q^n}\right)^2 \equiv 1+ 4\sum\_{n\geq 1} \frac{q^n}{1-q^{2n}}\pmod8.$$
For $k=2^s m$ ... | 14 | https://mathoverflow.net/users/7076 | 404708 | 165,968 |
https://mathoverflow.net/questions/404713 | 2 | [A question I remember from many years ago.]
Definition
>
> A **Riemann surface** is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point $x \in X$ there is ... | https://mathoverflow.net/users/454 | A Riemann surface is automatically paracompact | This is precisely the statement of [Radó's theorem](https://en.wikipedia.org/wiki/Rad%C3%B3%27s_theorem_(Riemann_surfaces)) (modulo the standard equivalence between paracompactness and second-countability for connected manifolds). I believe there are multiple proofs available, one of which is in section 1.3 of [Hubbard... | 5 | https://mathoverflow.net/users/30186 | 404714 | 165,969 |
https://mathoverflow.net/questions/404724 | 41 | Linear algebra as we learn it as undergraduates usually holds for any field (even though we usually learn it for the complex, or real, numbers).
I am looking for a list of concepts, and results, in linear algebra that actually depend on the choice of field.
To start I propose the notion of an complex valued inner p... | https://mathoverflow.net/users/167165 | Results in linear algebra that depend on the choice of field | The existence of Chevalley–Jordan decompositions depends on the perfectness of the field.
| 47 | https://mathoverflow.net/users/1409 | 404737 | 165,976 |
https://mathoverflow.net/questions/404705 | 4 | How can we from $\sum\_{n\leqslant x}\mu (n)=o(x)$ deduce $\sum\_{n\leqslant x}(-1)^{\omega(n)}=o(x)$, where $\omega(n)$ is the number of different primes dividing $n$ and
$\mu (n)$ is the Möbius function?
Heuristic: Up to x there must be roughly the same portion of numbers with odd and even number of prime divisors. F... | https://mathoverflow.net/users/169583 | Connection between Möbius and function | Yes, that deduction is possible, but somewhat indirectly.
The estimate $\sum\_{n \leq x} \mu(n) = o(x)$ is equivalent to the nonvanishing of $\zeta(s)$ on ${\rm Re}(s) = 1$ since both conditions are known to be equivalent to the prime number theorem. (Maybe one can directly deduce the estimate and the nonvanishing fr... | 9 | https://mathoverflow.net/users/3272 | 404743 | 165,978 |
https://mathoverflow.net/questions/404706 | 7 | For the sake of this question, we'll model a six functor formalism in the following way. Let $\mathsf{C}$ be a category *of spaces* (be it the category of schemes, or topological spaces) and consider a triangulated closed symmetric monoidal category $\mathsf{D}(X)$, with identity $\mathscr{O}\_X$, for each $X\in\mathsf... | https://mathoverflow.net/users/131975 | How duality follows from a six functor formalism | Your description of the six functors does not mention any relations between the $!$-functors and the $\*$-functors or the tensor product, which is where these dualities are hiding.
Poincaré duality is a relation between the $\*$-functors and the $!$-functors. Typically there are canonical isomorphisms $f\_!=f\_\*$ wh... | 11 | https://mathoverflow.net/users/20233 | 404750 | 165,980 |
https://mathoverflow.net/questions/404711 | 3 | I am looking at the following Riemann surface (let's call it $M$),
\begin{equation}
y^n=\frac{(x-x\_1)(x-x\_3)}{(x-x\_2)(x-x\_4)}
\end{equation}
which is a Riemann surface of genus $n-1$. It can be thought of as a quotient of the complex plane by a Schottky group $\Sigma$,
\begin{equation}
M\cong\mathbb{C}'/\Sigma
\end... | https://mathoverflow.net/users/351553 | Schwarzian derivative, accessory parameters, projective connections | First of all, you should take care about which coordinates you take. Namely, Faulkner uses the coordinate $x$ given by the defining equation $$y^n=\frac{(x-x\_1)(x-x\_2)}{(x-x\_3)(x-x\_4)}$$ as a branched projective structure (he denotes this by $z$ in his paper!). He then computes the Schwarzian derivative of $w$ - th... | 1 | https://mathoverflow.net/users/4572 | 404752 | 165,982 |
https://mathoverflow.net/questions/404761 | 2 | To generate a uniform distribution on a sphere $S^n$ in $\mathbb R^{n+1}$, we can normalize a vector whose entries are $n+1$ i.i.d normal random variables. If $\rho$ is a correlation, $|\rho|<1$, how can we generate a uniform distribution on the manifold
$$
M = \left \{ x, y \in S^n: x^Ty = \rho \right\}\ ?
$$
| https://mathoverflow.net/users/97209 | Uniform distribution on a manifold | It can be done in pretty much the same way as for a single vector by using the fact that if you fix $x$, then the conditional distribution of $y-x$ is uniform on the sphere of radius $\sqrt{1-\gamma^2}$ in the hyperplane perpendicular to $x$. Therefore, first you generate $x$ (as you say, by normalizing a vector with i... | 2 | https://mathoverflow.net/users/8588 | 404769 | 165,986 |
https://mathoverflow.net/questions/404760 | 20 | **Example:** How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this polynomial is not unique, because $2x^k$ and $x^{k+1}$ give the same value at $p=2$, so for example $2x^2+x+1$, $4x+x+1$... | https://mathoverflow.net/users/10446 | How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)? | You can certainly do better than brute force by considering modular constraints. If the solution is $p(x) = \sum\_i a\_i x^i$ then $p(x) - \sum\_{j=0}^{n-1} a\_j x^j$ is divisible by $x^n$ and $$\frac{p(x) - \sum\_{j=0}^{n-1} a\_j x^j}{x^n} = a\_n \pmod x$$
Solving for $a\_0$ in each of the given bases and using the ... | 26 | https://mathoverflow.net/users/46140 | 404772 | 165,988 |
https://mathoverflow.net/questions/404774 | 1 | Consider the [Young's lattice](https://en.wikipedia.org/wiki/Young%27s_lattice). *What is the number of paths starting from the origin (0) to a specific Young diagram?*
For instance, the Young diagram corresponding to the integer partition 1+1+1 has 1 path leading to it, 2+1 has 2 paths leading to it and 2+2 has 2 pa... | https://mathoverflow.net/users/140290 | Number of paths to a specific vertex in the Young's lattice | These paths are the same thing as standard Young tableaux, which are enumerated by the famous [Hook Length Formula](https://en.wikipedia.org/wiki/Hook_length_formula).
| 3 | https://mathoverflow.net/users/33089 | 404780 | 165,989 |
https://mathoverflow.net/questions/404786 | 2 | Naturally given any $s\in (0,1)$, the fractional Laplacian, $(-\Delta\_g)^s u$ on a closed Riemannian manifold can be defined via spectral decomposition of $-\Delta\_g$. There is another formulation of the fractional Laplacian that I commonly see that is stated as follows:
$$(-\Delta\_g)^sf(x)=\int\_0^{\infty} (e^{-t\D... | https://mathoverflow.net/users/50438 | Fractional Laplacian on closed manifolds | Yes, they are equivalent. Up to a constant missing and a sign error in the displayed formula, it should read:
$$ (-\Delta\_g)^s f(x) = \frac{1}{\Gamma(-s)} \int\_0^\infty (e^{t \Delta\_g} f(x) - f(x)) t^{-1-s} dt . $$
In fact, this is true for quite general operators. If $L^2$ theory is what you are after, this is a di... | 4 | https://mathoverflow.net/users/108637 | 404792 | 165,993 |
https://mathoverflow.net/questions/404805 | 3 | Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda\_1\lambda\_2\cdots\lambda\_r$ when $\lambda=(\lambda\_1\geq\lambda\_2\geq\cdots\geq\lambda\_r>0)$.
Let $\gamma$ be the Euler's constant. Lehmer proved that
$$\lim\_{n\rightarrow\infty}\frac1n\sum\_{\lambda\vdas... | https://mathoverflow.net/users/66131 | asymptotic growth of a sum involving partitions | The limit equals 2. We have $$
\sum\_{\lambda\vdash n}\frac1{\mathcal{N}(\lambda)^2}=[x^n]\prod\_{k=1}^\infty\frac1{1-x^k/k^2}=\sum\_{m=0}^n[x^m]\prod\_{k=2}^\infty\frac1{1-x^k/k^2}\\
=b\_0+b\_1+\ldots+b\_n,$$
where
$$
\sum b\_i x^i=\prod\_{k=2}^\infty \frac1{1-x^k/k^2}=:f(x),
$$
the standard uniform convergence argume... | 6 | https://mathoverflow.net/users/4312 | 404810 | 166,001 |
https://mathoverflow.net/questions/404777 | 1 | Let $B \ge 2$ be integer and $[x]$ denote the nearest integer
to real $x$.
For $2 \le B \le 10^5$ computations with mpmath suggest:
$$ [\zeta(1-1/B)]=-B+1 \qquad (1)$$
Is (1) true for all $B \ge 2$?
| https://mathoverflow.net/users/12481 | On the nearest integer to $\zeta(1-1/B),B \ge 2$ | We can make the error mentioned by Wojowu in his comment explicit by using some results on the Laurent coefficients of the zeta function. There are a few results on this, but I'll just use Theorem 2 of [this paper](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-2/issue-1/On-the-Hurwitz-... | 7 | https://mathoverflow.net/users/307675 | 404816 | 166,003 |
https://mathoverflow.net/questions/404803 | 2 | Let $D$ be a smooth domain of $\mathbb{R}^d$. Let $\partial D$ denote the boundary of $D$. We denote by $B(x,r)=\{y \in \mathbb{R}^d \mid |y-x|<r\}$ the Euclidean ball centered at $x$ with radius $r$
For $x \in \partial D$, we consider the following limit:
\begin{align\*}
\lim\_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int... | https://mathoverflow.net/users/68463 | On a characterization of inward unit normal vector | For small $r$ the curvature of the surface $\partial D$ can be neglected, so $D \cap B(x,r)$ is half the $d$-dimensional ball with radius $r$. Choosing the origin of the coordinate system at position $x$ and orienting the $x\_1$-axis along the inward normal, the integral is given by the vector $v$ with components
$$v\_... | 3 | https://mathoverflow.net/users/11260 | 404819 | 166,004 |
https://mathoverflow.net/questions/404806 | 2 | $\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$The compact simple Lie groups $\SO\_8(\mathbb{R}) $ and $\SO\_9(\mathbb{R}) $ both have rank 4. The group
$$
G=\SU\_3 \times \SU\_2 \times \operatorname{U}\_1
$$
also has rank 4. Does there exist a subgroup of $\SO\_8(\mathbb{R}) $ or $\SO\_9(\mathbb{R}) $ isomor... | https://mathoverflow.net/users/387190 | Smallest dimension for faithful orthogonal representation | I made some mistakes in my first version of this answer (including giving the opposite reply …), so hopefully this is correct. Thanks to @ZoltanZimboras for help sorting it out (although, of course, if it is still wrong, then the fault is entirely mine).
I claim that there is no embedding of $G$ in $\operatorname{SO}... | 2 | https://mathoverflow.net/users/2383 | 404831 | 166,008 |
https://mathoverflow.net/questions/404605 | 4 | This question is based on the assumption that $V \ne L$ and we have $\omega\_1^L < \omega\_1$ (here $\omega\_1^L$ is equal to the supremum of ordinals accidentally writable by no-oracle Ordinal Turing Machines).
Consider Ordinal Turing Machines (called “$\omega\_{\alpha}$-machines”) with an oracle that provides acces... | https://mathoverflow.net/users/122796 | How large are the stabilization times of Ordinal Turing Machines with an oracle for the transfinite initial ordinals? | For ordinal Turing machines with an oracle $S⊂Ord$,
- the set/class of output locations that are written at some point is $Σ^{L[S]}\_{1,S}$ (i.e. $Σ\_1$ in $L[S]$ with a predicate for $S$) and can be arbitrary such,
- the set/class of output locations that are eventually 1 is $Σ^{L[S]}\_{2,S}$ and can be arbitrar... | 3 | https://mathoverflow.net/users/113213 | 404838 | 166,014 |
https://mathoverflow.net/questions/404840 | -1 | Does there exist a non constant almost surely continuous stochastic process $X$ on $[0, \infty)$ with $X\_t$ independent of $X\_{t+1}$ for all $t \geq 0$?
| https://mathoverflow.net/users/173490 | A periodically independent stochastic process | Stupid answer: the trivial process $X\_t=0$.
Less stupid answer: for every half-integer $n/2$, choose $X\_{n/2}$ independently according to your favorite probability distribution and then interpolate linearly for other values of $t$.
| 2 | https://mathoverflow.net/users/47135 | 404842 | 166,015 |
https://mathoverflow.net/questions/404070 | 2 | Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a *quadratic category* with $(G,\tau,m)$ if its Grothendieck ring has basis $\{X\_g, Y\_g \ | \ g \in G\}$ and fusion rules (let $e$ be t... | https://mathoverflow.net/users/34538 | A twisted Haagerup category without pivotal structure | **Quick answer**: no.
Andrew Schopieray pointed out to me the PhD thesis of Josiah E. Thornton [2].
---
Let $\mathcal{F}$ be a fusion ring with basis $B=\{ b\_1, \dots, b\_r \}$. Let $G$ be the group of invertible elements $b\_i$ of $B$ (i.e. $\textrm{FPdim}(b\_i)=1$). The fusion ring $\mathcal{F}$ is called *q... | 1 | https://mathoverflow.net/users/34538 | 404843 | 166,016 |
https://mathoverflow.net/questions/404775 | 0 | Consider the function $\_2F\_1(5.5, 1, 5;-|x|^2]$ for $x\in \mathbb{R}^n.$ I want to show that this function is positive. I checked that it does not have any roots so can I conclude the inequality by using continuity in $x$ of the function $\_2F\_1(5.5, 1, 5;-|x|^2]$?
| https://mathoverflow.net/users/68232 | How to show the following inequality $_2F_1(5.5, 1, 5;-|x|^2]>0$? | You have the integral representation
$${}\_2F\_1\left(\begin{matrix}11/2,1\\5\end{matrix};-x\right)=\frac 4{x^4}\int\_0^x\frac{(x-t)^3}{(1+t)^{11/2}}\,dt,$$
which follows by expanding $1/(1+t)^{11/2}$ using the binomial theorem and integrating termwise. This should prove the positivity.
Note that this integral is a r... | 2 | https://mathoverflow.net/users/10846 | 404845 | 166,017 |
https://mathoverflow.net/questions/404782 | 2 | We will be working over an algebraically closed field of characteristic 0. We say that a projective variety $X\subset \mathbb{P}^n$ has projectively isomorphic plane sections if there is an open set $U\in(\mathbb{P}^n)^\vee$ such that the hyperplane sections $H\cap X,\ H\in U$ are all projectively isomorphic, i.e. for ... | https://mathoverflow.net/users/131868 | Does a smooth cubic in $P^3$ have projectively isomorphic sections? | Try this: Consider a "Lefschetz pencil" of planes through a general line. I believe these cut the smooth cubic surface in a pencil of irreducible plane cubic curves that are generally smooth, and the special ones have only one node, in particular are stable curves. Hence the smooth ones converge to the nodal ones in th... | 3 | https://mathoverflow.net/users/9449 | 404847 | 166,018 |
https://mathoverflow.net/questions/404817 | 17 | There are rare algebraic varieties such that the number of points over finite fields $\mathbb{F}\_p$ is given by a polynomial in $p$. One notable series of examples is the commuting variety: $[A,B]=0$ of $n\times n$ matrices $A,B$ over finite field. The computation was obtained by [Feit and Fine in 1960](https://projec... | https://mathoverflow.net/users/10446 | Number of $3\times 3$ anticommuting matrices over finite fields $\mathbb{F}_p$ is (or is not?) polynomial in $p$? | Let's work over a field $K$, which when finite is supposed to have $q$ elements. I'll assume the characteristic to be $\neq 2$, since in characteristic 2 we get the commuting variety which is well-known. Let $Q=Q(K)$ be the set of anticommuting pairs $(A,B)$.
If $A$ has eigenvalues $(x,y,z)$ (possibly on an extension... | 18 | https://mathoverflow.net/users/14094 | 404860 | 166,022 |
https://mathoverflow.net/questions/404830 | 5 | If $(a\_{n})$ is a conditionally convergent series in real field, then for any real number $\alpha$, there exists a rearrangement $(a\_{k\_{n}})$ of $(a\_{n})$ such that for all even $n$, $a\_{k\_{n}} \geq 0$, for all odd $n$, $a\_{k\_{n}} \leq 0$, and $\sum a\_{k\_{n}} = \alpha$.
This problem boils down to the follo... | https://mathoverflow.net/users/138147 | A restricted version of Riemann series theorem: rearrangements with alternating signs | Here you prescribe in addition the sequence of signs of the rearranged series in the Riemann-Dini theorem to be alternating, but note that any non-stationary binary sequence of signs does as well. More precisely:
>
> Let $(a\_k)\_{k\in\mathbb N} $ be an infinitesimal sequence of non-zero real
> numbers such that $\... | 4 | https://mathoverflow.net/users/6101 | 404861 | 166,023 |
https://mathoverflow.net/questions/404844 | 2 | I wonder whether one can exactly calculate the following integral in terms of $d$ and $p\geq 1$ or not, or a better bound(than the trivial one I am going to give) in terms of $d,p$:
$$\left(\int\_{[0,1)^d}\|x\|\_2^p\,dx\right)^{1/p},$$
where $\|x\|\_2$ is the Euclidean distance from $x$ to 0. Trivially, one has $\|... | https://mathoverflow.net/users/174600 | Exact formula or non-trivial upper bound on p-norm of $f(x)=\|x\|_2$ in $[0,1)^d$ | $\DeclareMathOperator\E{E}\DeclareMathOperator\Var{Var}\DeclareMathOperator\P{P}$Note that
$$\int\_{[0,1)^d}\|x\|\_2^p\,dx
=\E S\_d^{p/2},\tag{1}\label{1}$$
where $S\_d:=\sum\_1^d U\_j^2$ and the $U\_j$'s are iid random variables uniformly distributed on the interval $[0,1]$.
Note next that $\E S\_d=d/3$ and $\Var... | 3 | https://mathoverflow.net/users/36721 | 404864 | 166,025 |
https://mathoverflow.net/questions/404863 | 1 | The expectation values of the [1D simple random walk](https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk) $S\_n$ can be [shown](https://mathworld.wolfram.com/RandomWalk1-Dimensional.html) to have the asymptotic behavior of
$$ \lim\_{n\to\infty} \frac{a\_n}{n^{1/2}} = \sqrt{\frac{2}{\pi}}, \tag{1}\lab... | https://mathoverflow.net/users/83999 | Asymptotics of cumulative Liouville function under RH versus simple random walk | I figured the remarks I gave in the comments deserve to be gathered up into a more coherent form as an answer.
One thing I will start with is that comparing $L(n)$ (or $M(n)$, the Mertens function) to the values $S\_n$ is arguably not the right heuristic. Indeed, $S\_n$ averages over all random walks, which can have ... | 3 | https://mathoverflow.net/users/30186 | 404868 | 166,029 |
https://mathoverflow.net/questions/404853 | 2 | Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module.
A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-module complement. But this is a very strong assumption. What are weaker but sufficient criteria for an $R$-module comple... | https://mathoverflow.net/users/167165 | Module complements to rings embedded in a projective module | For (A) a ring $R$ is selfinjective if it is injective as an $R$-module (on e should say left or right). By definition of injective this means $R$ has a complement in any module. Examples include Frobenius and quasi-frobenius rings.
If $R$ is von Neumann regular, then any finitely generated submodule of a projective ... | 2 | https://mathoverflow.net/users/15934 | 404871 | 166,031 |
https://mathoverflow.net/questions/404880 | 2 | Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x\_1+y\_1, \dots,
x\_n+y\_n) \leq f(x\_1, \dots, x\_n) + f(y\_1, \dots, y\_n)$ in its domain, $f \geq 0$, and $f$ is nondecreasing in $x$ and nonincreasi... | https://mathoverflow.net/users/43628 | Superhomogeneity of subadditive functions | The answer is no. E.g., let $f(0):=0$ and $f(x):=1+m\_+$ if $|x|\in[2^{m-1},2^m)$ for $x\in\mathcal C:=[0,\infty)^n$ and an integer $m$; that is, $f(x)=1+(1+\lfloor\log\_2|x|\rfloor)\_+$ for all $x\in\mathcal C\setminus\{0\}$. Here, $m\_+:=\max(0,m)$ and $|x|$ is the Eucludean norm of $x$.
**Details:** Clearly, $f$ i... | 2 | https://mathoverflow.net/users/36721 | 404883 | 166,035 |
https://mathoverflow.net/questions/404876 | 1 | Consider a separable Hilbert space $\mathcal H$ and the bounded linear operators $B(\mathcal H)$.
Consider a function $T: [0, \infty) \to B(\mathcal H)$, under what assumptions on $T(t)$ is it true that
$$\big(\int\_0^c T(t) \, dt \big) (x) = \int\_0^c T(t)x \, dt \ , \ \ \ \forall x \in \mathcal H , c \in (0, \infty)$... | https://mathoverflow.net/users/161092 | Action of Bochner integral of operator-valued functions on vectors | I understand you assume that $T:[0,c]\to\mathcal{B(H)}$ is Bochner integrable in order to write $\int\_0^c T(t)dt$ as Bochner integral. Then for any $x\in\mathcal H$ the map $ [0,c]\ni t\mapsto \mathcal H$ is also Bochner integrable and the identity you wrote holds. More generally: for a measure situation $(X,\mathcal ... | 4 | https://mathoverflow.net/users/6101 | 404889 | 166,037 |
https://mathoverflow.net/questions/404626 | 3 | It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that
$$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$
In particular, if you take a $3$-dimensional manifold with boundary given by a genus $g$ surface, then its Euler characteristic is $\... | https://mathoverflow.net/users/259525 | Euler characteristic of pseudomanifolds with boundary | Ok, to convert my comment to an answer. Let $S$ be a closed orientable triangulated surface of genus $\ge 1$. Let $M$ be the cone over $S$. Then $M$ has a natural orientable pseudomanifold structure. However, $\chi(M)=1$, while $\chi(S)$ can be any nonpositive even number.
The moral is that there are way too many pse... | 4 | https://mathoverflow.net/users/39654 | 404892 | 166,039 |
https://mathoverflow.net/questions/404852 | 3 | A proper edge $k$-coloring of a graph is an assignment of $k$ colors to the edges of the graph so that no two adjacent edges have the same color. The smallest integer $k$ such that $G$ has a proper edge $k$-coloring is the chromatic index of $G$.
Giving a simple graph $G$, the well-known Vizing's theorem tells us tha... | https://mathoverflow.net/users/375270 | An edge coloring problem for class two graphs | Yes!
Let $uv$ be an edge colored by the last color, say $\Delta+1$. If $uv$ is incidence with all colors, then it is the required edge. So the only case is that every edge colored with $\Delta+1$ is not incident with some color in $[\Delta]$, and thus it can be recolored by that color. This results an edge $\Delta$-c... | 5 | https://mathoverflow.net/users/148974 | 404928 | 166,060 |
https://mathoverflow.net/questions/404778 | 4 | Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:S\to S$ with $LG=1$. Retarded means that for every $g\in S$ whose support is disjoint from the past causal cone of $x$, ... | https://mathoverflow.net/users/56920 | spaces of smooth functions for linear hyperbolic PDE | For normally hyperbolic operators (those whose principal symbol is the same as for the wave operator, but possibly acting on a vector bundle, the theory of fundamental solutions/Green functions (as distributions that would be acting on smooth functions) is very well developed in
>
> *Bär, Christian; Ginoux, Nicolas... | 5 | https://mathoverflow.net/users/2622 | 404934 | 166,063 |
https://mathoverflow.net/questions/404875 | 2 | This is a follow-up question to [Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$](https://mathoverflow.net/questions/403542/positive-integer-solutions-to-the-diophantine-equation-xz1yz1-z4z3-z/404278#404278)
Let \begin{equation}
P(x,n)= 1+x+x^2+ \cdots + x^n, \end{equation}
... | https://mathoverflow.net/users/166404 | Positive divisors of $P(x,n)=1+x+x^2+ \cdots + x^n$ that are congruent to $1$ modulo $x$ | Conjecture 1 follows from a (multiple of) [Bezout identity](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity) for polynomials:
$$mu(x)f(x)+mv(x)g(x)=md(x),$$
where $d(x) :=\gcd(f(x),g(x))$ and a positive integer $m$ is chosen such that polynomials $mu(x)$, $mv(x)$, $md(x)$ have integer coefficients. Then $f(x)\mi... | 6 | https://mathoverflow.net/users/7076 | 404937 | 166,064 |
https://mathoverflow.net/questions/404942 | 1 | I asked the following question in a forum more suitable for statistics, but I didn't get any answer; I hope, someone could shed light on my question:
I have three random variables, $X\_1$, $X\_2$, and $X\_3$, which they are distributed normally. If we consider an estimator which reads as:
$$\hat{\theta} = a\_1 X\_1 +... | https://mathoverflow.net/users/nan | Jeffreys' priors as coefficients of a linear estimator | The coefficients $a\_i$ of the random variables $X\_i$ are not any prior probabilities at all -- because prior probabilities are coefficients, not of random variables, but of probability distributions.
The choice $a\_i\propto 1/\sigma\_i$ in your setting equalizes the variances of the random variables $a\_iX\_i$, and... | 1 | https://mathoverflow.net/users/36721 | 404943 | 166,066 |
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