An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group G by $\psi (G)$ and let $C_n$ denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q?1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, k odd, there exists a group G of order n satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if G is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of G.     

Two refined major-balance identities on 321-avoiding involutions

Making use of a combinatorial approach, we prove two refined major-balance identities on the $321$-avoiding involutions in ${\mathfrak{S}}_n$, respecting the number of fixed points and the number of descents, respectively. The former one is proved in terms of ordered trees whose non-root nodes have exactly two children, and the latter one is proved in terms of lattice paths within a $\lfloor \frac{n}{2}\rfloor \times \lceil \frac{n}{2}\rceil$ rectangle.     

A proof of Wright's conjecture

Wright's conjecture states that the origin is the global attractor for the delay differential equation $y^{\prime }(t)=?\alpha y(t?1)[1+y(t)]$ for all $\alpha \in (0,\frac{\pi }{2}]$ when $y(t)>?1$. This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range $\alpha \in (0,\frac{\pi }{2}]$, and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha =\frac{\pi }{2}$. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for $\alpha \in (\frac{\pi }{2},\frac{\pi }{2}+6.830\times {10}^{?3}]$. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at $\alpha =\frac{\pi }{2}$ is globally parametrized by $\alpha >\frac{\pi }{2}$.     

On a conjecture of Füredi

Füredi conjectured that the Boolean lattice $2^{\left[n\right]}$ can be partitioned into $\left(\genfrac{}{}{00}{}{n}{\lfloor n/2\rfloor }\right)$ chains such that the size of any two differs in at most one. In this paper, we prove that there is an absolute constant $\alpha \approx 0.8482$ with the following property: for every $ϵ>0$, if $n$ is sufficiently large, the Boolean lattice $2^{\left[n\right]}$ has a chain partition into $\left(\genfrac{}{}{00}{}{n}{\lfloor n/2\rfloor }\right)$ chains, each of them of size between $(\alpha -ϵ)\sqrt{n}$ and $O(\sqrt{n}/ϵ)$.We deduce this result by looking at the more general setup of unimodal normalized matching posets. We prove that a unimodal normalized matching poset $P$ of width $w$ has a chain partition into $w$ chains, each of size at most $\frac{2\left|P\right|}{w}+5$, and it has a chain partition into $w$ chains, where each chain has size at least $\frac{\left|P\right|}{2w}-\frac{1}{2}$.     

Corrigendum to “Scattering above energy norm of solutions of a loglog energy-supercritical Schrödinger equation with radial data”

The purpose of this corrigendum is to point out some errors that appear in [1]. Our main result remains valid, i.e scattering of ${\stackrel{?}{H}}^k:={\dot{H}}^k({\mathbb{R}}^n)\cap {\dot{H}}^1({\mathbb{R}}^n)$ solutions of the loglog energy-supercritical Schrödinger equation $i{?}_{t}u+△u=|u{|}^{\frac{4}{n?2}}u{\log }^c?(\log ?(10+|u{|}^2)$, $0<c<c_n$, $n\in \{3,4\}$, with $k>\frac{n}{2}$, radial data $u(0):=u_0\in {\stackrel{?}{H}}^k$ but with slightly different values of $c_n$, i.e $c_n=\frac{1}{5772}$ if $n=3$ and $c_n=\frac{3}{8024}$ if $n=4$. We propose some corrections.