Large blocking sets in PG(2,q2)

Minimal blocking sets in $\mathrm{PG}(2,q^2)$ have size at most $q^3+1$. This result is due to Bruen and Thas and the bound is sharp, sets attaining this bound are called unitals. In this paper, we show that the second largest minimal blocking sets have size at most $q^3+1-(p-3)/2$, if $q=p$, $p\ge 67$, or $q=p^h$, $p>7$, $h>1$. Our proof also works for sets having at least one tangent at each of its points (that is, for tangency sets).     

Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic $p\ge 3$, in particular their standard forms $C:x^4+y^4+z^4+r{x}^2{y}^2+s{y}^2{z}^2+t{z}^2{x}^2=0$. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then $r,s,t$ belong to ${\mathbb{F}}_{p^2}$ and C is maximal or minimal over ${\mathbb{F}}_{p^2}$. Moreover, in this case we will provide a simple criterion in terms of $r,s,t,p$ that tells whether C is maximal (resp. minimal) over ${\mathbb{F}}_{p^2}$.     

On the inhomogeneous biharmonic nonlinear Schrödinger equation: Local,global and stability results

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $i{u}_t+{\Delta }^{2}u+\lambda {\left|x\right|}^{-b}{\left|u\right|}^{\alpha }u=0,$ where $\lambda =\pm 1$ and $\alpha$, $b>0$. We show local and global well-posedness in $H^s\left({\mathbb{R}}^N\right)$ in the $H^s$-subcritical case, with $s=0,2$. Moreover, we prove a stability result in $H^2\left({\mathbb{R}}^N\right)$, in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates related to the linear problem.     

Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

Some new results on dimension and Bose distance for various classes of BCH codes

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over ${\mathbb{F}}_q$ with designed distance $\delta =a{q}^{m-1}-1(\text{resp. }\delta =a\frac{q^m-1}{q-1})$ for all $1\le a\le q-1$, where q is a prime power and $m>1$ is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16], and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range $[a\frac{q^m-1}{q-1},a\frac{q^m-1}{q-1}+T]$ for $0\le a\le q-2$, where $T=q^{\frac{m+1}{2}}-1$ if m is odd, and $T=2q^{\frac{m}{2}}-1$ if m is even.     

A new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in terms of the density $\rho$ and the pressure $P$. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm $\left|\right|\rho {\left|\right|}_{L^{\infty }(0,t;L^{\infty })}+\left|\right|P{\left|\right|}_{L^{p_0}(0,t;L^{\infty })}<\infty$ for some constant  $p_0$ satisfying $1<p_0\le 2$. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as $\left|\right|\rho {\left|\right|}_{L^{\infty }(0,t;L^{\infty })}+\left|\right|P{\left|\right|}_{L^{\infty }(0,t;L^{\infty })}<\infty$.     

Classification of some quadrinomials over finite fields of odd characteristic

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, is a permutation quadrinomial of ${\mathbb{F}}_{q^2}$ over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where $char({\mathbb{F}}_q)=2$ and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial $x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $char({\mathbb{F}}_q)=3,5$ and $a,b,c\in {\mathbb{F}}_q^{⁎}$ and proposed some new classes of permutation quadrinomials of ${\mathbb{F}}_{q^2}$.In particular, in this paper we classify all permutation polynomials of ${\mathbb{F}}_{q^2}$ of the form $f(x)=x^3+a{x}^{q+2}+b{x}^{2q+1}+c{x}^{3q}$, where $a,b,c\in {\mathbb{F}}_q^{⁎}$, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.     

Intersections between the norm-trace curve and some low degree curves

In this paper we analyze the intersection between the norm-trace curve over ${\mathbb{F}}_{q^3}$ and the curves of the form $y=a{x}^3+b{x}^2+cx+d$, giving a complete characterization of the intersection between the curve and the parabolas (a=0), as well as sharp bounds for the other cases. This information is used for the determination of the weight distribution of some one-point AG codes arising from the curve.     

Dynamical behavior of solutions of a reaction–diffusion model in river network

In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant $\beta$. We show the existence of two critical values $c_0$ and 2 with $0<c_0<2$, and prove that when $-c_0\le \beta <2$, the population density in every branch of the river goes to 1 as time goes to infinity; when $-2<\beta <-c_0$, then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when $\left|\beta \right|\ge 2$, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., $\left|\beta \right|<2$), the species will survive in the long run.