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.     

Boundary layers for a fluid–particle interaction system with density-dependent viscosity and cylindrical symmetry

In this paper, we study the initial boundary value problem for a cylindrical symmetry fluid–particle interaction system in three dimensions. The boundary layer phenomena is investigated when the shear viscosity $\mu =\kappa {\rho }^{\beta }$ goes to zero. Furthermore, we establish the boundary layer thickness of the order $O\left({\kappa }^{\alpha }\right)$ for more general initial data when $0<\alpha <\frac{1}{2}$ and give the optimal boundary-layer thickness for the system with more general initial data. As a byproduct, this work improves the corresponding results in Yao et al. (2011) for isentropic compressible Navier–Stokes equations where $0<\alpha <\frac{1}{4}$.     

Remarks on the distribution of colors in Gallai colorings

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \cdots \ge e_k$ of positive integers is an $(n,k)$-sequence if ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i\le k$. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer $k\ge 3$ there exists an integer $g\left(k\right)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g\left(k\right)$. They showed that $g\left(3\right)=5,g\left(4\right)=8$ and $2k-2\le g\left(k\right)\le 8k^2+1$.We show that $g\left(5\right)=10$ and give almost matching lower and upper bounds for $g\left(k\right)$ by showing that with suitable constants $\alpha ,\beta >0$, $\frac{\alpha k^{1.5}}{lnk}\le g\left(k\right)\le \beta k^{1.5}$ for all sufficiently large $k$.     

Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

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.     

LP-based approximation for uniform capacitated facility location problem

In this paper, we study uniform hard capacitated facility location problem. The standard LP for the problem is known to have an unbounded integrality gap. We present constant factor approximation by rounding a solution to the standard LP with a slight $(1+ϵ)$ violation in the capacities.Our result shows that the standard LP is not too bad.Our algorithm is simple and more efficient as compared to the strengthened LP-based true approximation that uses the inefficient ellipsoid method with a separation oracle. True approximations are also known for the problem using local search techniques that suffer from the problem of convergence. Moreover, solutions based on standard LP are easier to integrate with other LP-based algorithms.The result is also extended to give the first approximation for uniform hard capacitated $k$-facility location problem violating the capacities by a factor of $(1+ϵ)$ and breaking the barrier of 2 in capacity violation. The result violates the cardinality by a factor of $\frac{2}{1+ϵ}$.     

The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution $\theta =|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}{ϵ}$ to a time scale of $\frac{1}{{ϵ}^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.     

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.     

Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $[0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $[0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.     

Erdős matching conjecture for almost perfect matchings

In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^3$ and $(s+1)k?n<(s+1)(k+\frac{1}{100k})$.     

Note on rainbow cycles in edge-colored graphs

Let G be a graph of order n with an edge-coloring c, and let ${\delta }^c(G)$ denote the minimum color-degree of G. A subgraph F of G is called rainbow if all edges of F have pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if ${\delta }^c(G)>\frac{2n?1}{3}$, then every vertex of G is contained in a rainbow triangle; (ii) if ${\delta }^c(G)>\frac{2n?1}{3}$ and $n\ge 13$, then every vertex of G is contained in a rainbow $C_4$; (iii) if G is complete, $n\ge 7k?17$ and ${\delta }^c(G)>\frac{n?1}{2}+k$, then G contains a rainbow cycle of length at least k, where $k\ge 5$.