Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term

This paper discusses the quasilinear Schrödinger equation $-\Delta u+V\left(x\right)u-\Delta \left[{(1+u^2)}^{\frac{1}{2}}\right]\frac{u}{2{(1+u^2)}^{\frac{1}{2}}}=K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N,$where $N⩾3$. Under appropriate assumptions on the potentials $V$ and $K$ and local sublinear growth assumptions on the nonlinear term $f$, we get the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution.     

Asymptotically good ZprZps-additive cyclic codes

We construct a class of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic Gilbert-Varshamov bound at $\frac{1+p^{s-r}}{2}\delta$ is greater than $\frac{1}{2}$ and the relative distance of the code is convergent to δ, while the rate is convergent to $\frac{1}{1+p^{s-r}}$ for $0<\delta <\frac{1}{1+p^{s-r}}$ and $1\le r<s$. As a consequence, we prove that there exist numerous asymptotically good ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes.     

The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity ${\mathrm{e}}^{\frac{2\pi \mathbf{i}}{k}}$ (and its symmetric entry is ${\mathrm{e}}^{?\frac{2\pi \mathbf{i}}{k}}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set $\{1,{\mathrm{e}}^{\frac{2\mathit{\pi }\mathbf{i}}{k}},{\mathrm{e}}^{?\frac{2\pi \mathbf{i}}{k}}\}$, which is not closed under multiplication when $k?4$. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank ($H_k$-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with $H_k$-rank 2 and 3, respectively.     

Moser–Trudinger inequality involving the anisotropic Dirichlet norm (∫ΩFN(∇u)dx)1N on W01,N(Ω)

We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm ${({\int }_{\mathrm{\Omega }}{F}^N(\mathrm{?}u)dx)}^{\frac{1}{N}}$ on $W_0^{1,N}(\mathrm{\Omega })$ for $N\ge 2$. Here F is convex and homogeneous of degree 1, and its polar $F^o$ represents a Finsler metric on ${\mathbb{R}}^N$. Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.     

Panchromatic 3-colorings of random hypergraphs

The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let $H(n,k,p)$ denote the binomial model of a random $k$-uniform hypergraph on $n$ vertices. For given fixed $c>0$, $k⩾3$ and $p=cn∕\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, we prove that if $c<\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}-O\left({\left(\frac{\sqrt{3}}{2}\right)}^k\right)$then $H(n,k,p)$ admits a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$, but if $k$ is large enough and $c>\frac{ln3}{3}\cdot {\left(\frac{3}{2}\right)}^k-\frac{ln3}{2}+O\left({\left(\frac{3}{4}\right)}^k\right)$then $H(n,k,p)$ does not admit a panchromatic 3-coloring with probability tending to 1 as $n\to \infty$.     

On eigenfunctions and maximal cliques of generalised Paley graphs of square order

Let GP $(q^2,m)$ be the m-Paley graph defined on the finite field with order $q^2$. We study eigenfunctions and maximal cliques in generalised Paley graphs GP $(q^2,m)$, where $m|(q+1)$. In particular, we explicitly construct maximal cliques of size $\frac{q+1}{m}$ or $\frac{q+1}{m}+1$ in GP $(q^2,m)$, and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue $-\frac{q+1}{m}$ of GP $(q^2,m)$. These new results extend the work of Baker et al. and Goryainov et al. on Paley graphs of square order. We also study the stability of the Erdős-Ko-Rado theorem for GP $(q^2,m)$ (first proved by Sziklai).     

The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation

The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δu and the magnetic diffusion Δb by $?{(?\mathrm{\Delta })}^{\alpha }u$ and $?{(?\mathrm{\Delta })}^{\beta }b$, respectively, the resulting equations with $\alpha \ge \frac{5}{4}$ and $\alpha +\beta \ge \frac{5}{2}$ then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion $?{(?{\mathrm{\Delta }}_h)}^{\frac{5}{4}}b$, where ${\mathrm{\Delta }}_h={?}_1^2+{?}_2^2$.