On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in Rn

In this note, we mainly study the relation between the sign of ${(?\mathrm{\Delta })}^{p}u$ and ${(?\mathrm{\Delta })}^{p?i}u$ in ${\mathbb{R}}^n$ with $p?2$ and $n?2$ for $1?i?p?1$. Given the differential inequality ${(?\mathrm{\Delta })}^{p}u<0$, first we provide several sufficient conditions so that ${(?\mathrm{\Delta })}^{p?1}u<0$ holds. Then we provide conditions such that ${(?\mathrm{\Delta })}^{i}u<0$ for all $i=1,2,\dots ,p?1$, which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(?\mathrm{\Delta })}^{p}u={\mathrm{e}}^{2pu}$ and ${(?\mathrm{\Delta })}^{p}u=u^q$ with $q>0$ in ${\mathbb{R}}^n$.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

On even (unitary) perfect polynomials over F2

We give all even perfect (resp. unitary perfect) polynomials over the prime field ${\mathbb{F}}_2$ of the form $x^a{(x+1)}^b{M}_1^{h_1}?M_r^{h_r}$, where each $M_i$ is a Mersenne irreducible polynomial, $h_i=2^{n_i}?1$ (resp. $h_i=2^{n_i}$) and $a,b,r,n_i\in \mathbb{N}$. In particular, we characterize nine of the eleven known “sporadic” even perfect polynomials over ${\mathbb{F}}_2$ that have the above form.     

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O\left(k^5\right)}?n^9$. In this article we improve this running time to $2^{O\left(k^2\right)}?n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O\left(k^5\right)$ to $O\left(k^2\right)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p?1$ for $p\ge 3$, we speed up the running time to $2^{O(k?k^{\frac{1}{p?1}})}?n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p\ge 3$, Weighted Independent Set cannot be solved in time $2^{o\left(k\right)}?n^{O\left(1\right)}$ in the class of $\{\text{bull},C_4,\dots ,C_{2p?1}\}$-free graphs unless the ETH fails.     

On hypergraphs without loose cycles

Recently, Mubayi and Wang showed that for $r\ge 4$ and $?\ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $?$ is at most $2^{O\left(n^{r?1}{(logn)}^{(r?3)∕(r?2)}\right)}$. We improve this bound to $2^{O(n^{r?1}loglogn)}$.     

Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra

We define a family ${\mathrm{KV}}^{(g,n+1)}$ of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with $n+1$ boundary components. The problem ${\mathrm{KV}}^{(0,3)}$ is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to ${\mathrm{KV}}^{(g,n+1)}$ for arbitrary g and n. The key point is the solution to ${\mathrm{KV}}^{(1,1)}$ based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra ${\mathfrak{g}}^{(g,n+1)}$. In more detail, we show that every solution to ${\mathrm{KV}}^{(g,n+1)}$ induces a Lie bialgebra isomorphism between ${\mathfrak{g}}^{(g,n+1)}$ and its associated graded $\mathrm{gr}{\mathfrak{g}}^{(g,n+1)}$. For $g=0$, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For $g\ge 1$, $n=0$, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.