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 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 $2^{O(n^{r?1}loglogn)}$.     

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

3-Restricted arc connectivity of digraphs

The $k$-restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset $S$ of a strong digraph $D$ is a $k$-restricted arc cut if $D?S$ has a strong component $D^{\prime }$ with order at least $k$ such that $D?V\left(D^{\prime }\right)$ contains a connected subdigraph with order at least $k$. The $k$-restricted arc connectivity ${\lambda }^k\left(D\right)$ of a digraph $D$ is the minimum cardinality over all $k$-restricted arc cuts of $D$.Let $D$ be a strong digraph with order $n\ge 6$ and minimum degree $\delta \left(D\right)$. In this paper, we first show that ${\lambda }^3\left(D\right)$ exists if $\delta \left(D\right)\ge 3$ and, furthermore, ${\lambda }^3\left(D\right)\le {\xi }^3\left(D\right)$ if $\delta \left(D\right)\ge 4$, where ${\xi }^3\left(D\right)$ is the minimum 3-degree of $D$. Next, we prove that ${\lambda }^3\left(D\right)={\xi }^3\left(D\right)$ if $\delta \left(D\right)\ge \frac{n+3}{2}$. Finally, we give examples showing that these results are best possible in some sense.     

A sharp Balian–Low uncertainty principle for shift-invariant spaces

A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators ${\{f_k\}}_{k=1}^K?L^2({\mathbb{R}}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy ${\int }_{{\mathbb{R}}^d}|x||f_k(x){|}^{2}dx=\infty$, namely, $\stackrel{?}{f_k}?H^{1/2}({\mathbb{R}}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+?}({\mathbb{R}}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}({\mathbb{R}}^d)$. Our results are sharp in the sense that $H^{1/2}({\mathbb{R}}^d)$ cannot be replaced by $H^s({\mathbb{R}}^d)$ for any $s<1/2$.     

Note of the enumeration formula of standard Young tableaux of truncated shape (nm)∖(2)

In this paper integration method is used to derive the enumeration formulas of the standard Young tableau (SYT) of truncated shape. An integration representation of the number of SYT of truncated shape $\left(n^m\right)\setminus \left(2\right)$ is given, which implies a positive proof of the conjecture of Adin et al. on the number of SYT of truncated shape $\left(n^n\right)\setminus \left(2\right)$. Furthermore, the enumeration formulas of the numbers of SYTs of truncated shapes $\left(n^{n-k}\right)\setminus \left(2\right)$ and $\left(n^{n-k}\right)\setminus (1,1)$$(k=1,2)$ have been obtained.