Non-Wieferich primes under the abc conjecture

Assuming the abc conjecture, Silverman proved that, for any given positive integer $a?2$, there are $?\log ?x$ primes $p?x$ such that $a^{p?1}?1(\mathrm{mod}{p}^2)$. In this paper, we show that, for any given integers $a?2$ and $k?2$, there still are $?\log ?x$ primes $p?x$ satisfying $a^{p?1}?1(\mathrm{mod}{p}^2)$ and $p\equiv 1(\mathrm{mod}k)$, under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.     

Nash multiplicities and isolated points of maximum multiplicity

Let X be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$ be a point in the closed subset of maximum multiplicity of X. We provide a criterion, given in terms of arcs, to determine whether ξ is isolated in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$. More precisely, we use invariants of arcs derived from the Nash multiplicity sequence to characterize when ξ is an isolated point in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$.     

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations$\{\begin{array}{cc}{\mathrm{\Delta }}^{2}u?\mathrm{\Delta }u+V(x)u?\frac{1}{2}u\mathrm{\Delta }(u^2)=g(u),\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in H^2({\mathbb{R}}^N)\hfill \end{array}$ where ${\mathrm{\Delta }}^2=\mathrm{\Delta }(\mathrm{\Delta })$ is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition $g(t)t\ge \mu G(t)\ge 0$ with $\mu >3$. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.     

Derived recollements and generalised AR formulas

The Defect Recollement, Restriction Recollement, Auslander–Gruson–Jensen Recollement, and others, are shown to be instances of a general construction using zeroth derived functors and methods from stable module theory. The right derived functors ${\mathsf{W}}_k:=R_k{(\underset{\_}{})}^{?}$ are computed and it is shown that the functor ${\mathsf{W}}_2:=R_2{(\underset{\_}{})}^{?}$ is right exact and restricts to a duality $\mathsf{W}$ of the defect zero functors. The duality $\mathsf{W}$ satisfies two identities which we call the Generalised Auslander–Reiten formulas. We show that $\mathsf{W}$ induces the generalised Auslander–Bridger transpose and show that the Generalised Auslander–Reiten formulas reduce to the well-known Auslander–Reiten formulas.     

The spectrum of resolvable holey Mendelsohn triple systems and holey Mendelsohn frames with block size three

A holey Mendelsohn triple system (HMTS) is a decomposition of a complete multipartite directed graph into directed cycles of length 3. If the directed cycles of length 3 can be partitioned into parallel classes, then the HMTS is called an RHMTS. Bennett, Wei and Zhu [J. Combin. Des., 1997] showed that an RHMTS of type $g^n$ exists when $gn\equiv 0(\mathrm{mod}3)$ and $(g,n)\ne (1,6)$ with some possible exceptions. In this paper, motivated by the application in constructing RHMTSs, we investigate the constructions of holey Mendelsohn frames. We prove that a 3-MHF of type $(n,h^t)$ exists if and only if $n\ge 3$, $t\ge 4$ and $nh(t-1)\equiv 0(\mathrm{mod}3)$, and then determine that the necessary condition for the existence of an RHMTS of type $g^n$, namely, $gn\equiv 0(\mathrm{mod}3)$ is also sufficient except for $(g,n)=(1,6)$. New recursive constructions on incomplete RHMTSs via MHFs are introduced to settle this problem completely.     

Exact decay estimate on solutions of critical p-Laplacian equations in R+N

Consider the critical p-Laplacian equation in ${\mathbb{R}}_{+}^N$

$\{\begin{array}{cc}\hfill & ?{\mathrm{\Delta }}_{p}u=0,\text{in}{\mathbb{R}}_{+}^N,\hfill \\ \hfill & |\mathrm{?}u{|}^{p?2}\frac{?u}{?n}=|u{|}^{\stackrel{￣}{p}?2}u,\text{on}{\mathbb{R}}^{N?1},\hfill \end{array}$

where ${\mathbb{R}}_{+}^N={\mathbb{R}}^{N?1}\times (0,\infty )$, $1<p<N,\stackrel{￣}{p}=\frac{(N?1)p}{N?p}$ is the critical exponent of the Sobolev imbedding from $W^{1,p}({\mathbb{R}}_{+}^N)$ to $L^q({\mathbb{R}}^{N?1})$, $p\le q\le \stackrel{￣}{p}$ and ${\mathrm{\Delta }}_p$ is the p-Laplacian operator, ${\mathrm{\Delta }}_{p}u=\mathrm{?}(|\mathrm{?}u{|}^{p?2}\mathrm{?}u)$. We prove polynomial decay of the solutions

$|u(x)|\le c{(1+|x|)}^{?\frac{N?p}{p?1}},\text{for}x\in {\mathbb{R}}_{+}^N.$

The decay exponent is the best possible.     

Lorentz estimates for a class of nonlinear parabolic systems

In this paper we obtain the following local Lorentz estimates

$B\left(\left|F\right|\right)\in L_{loc}^{\gamma ,q}?B\left(\left|\mathrm{?}u\right|\right)\in L_{loc}^{\gamma ,q}\text{for any}\gamma >1\text{and}0<q\le \infty$

of the weak solutions for a class of quasilinear parabolic systems

$u_t?\text{div}\left(a\left(\left|\mathrm{?}u\right|\right)\mathrm{?}u\right)=\text{div}\left(a\left(\left|F\right|\right)F\right),$

where $B(t)={\int }_0^t\tau a(\tau )d\tau$ for $t\ge 0$.