On the number of solutions of certain diagonal equations over finite fields

We use character sums over finite fields to give formulas for the number of solutions of certain diagonal equations of the form$a_1{x}_1^{m_1}+a_2{x}_2^{m_2}+\cdots +a_n{x}_n^{m_n}=c.$ We also show that if the value distribution of character sums ${\sum }_{x\in {\mathbb{F}}_q}\chi (a{x}^m+bx)$, $a,b\in {\mathbb{F}}_q$, is known, then one can obtain the number of solutions of the system of equations$\{\begin{array}{cc}\hfill x_1+x_2+\cdots +x_n\hfill & \hfill =\alpha \hfill \\ \hfill x_1^m+x_2^m+\cdots +x_n^m\hfill & \hfill =\beta ,\hfill \end{array}$ for some particular m. We finally apply our results to induce some facts about Waring's problems and the covering radius of certain cyclic codes.     

3-dynamic coloring of planar triangulations

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that any vertex $v$ has at least $min\{r,{deg}_G\left(v\right)\}$ distinct colors in $N_G\left(v\right)$. The $r$-dynamic chromatic number${\chi }_r^d\left(G\right)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$.Loeb et al. (2018) showed that if $G$ is a planar graph, then ${\chi }_3^d\left(G\right)\le 10$, and there is a planar graph $G$ with ${\chi }_3^d\left(G\right)=7$. Thus, finding an optimal upper bound on ${\chi }_3^d\left(G\right)$ for a planar graph $G$ is a natural interesting problem. In this paper, we show that ${\chi }_3^d\left(G\right)\le 5$ if $G$ is a planar triangulation. The upper bound is sharp.     

Three new results on continuation criteria for the 3D relativistic Vlasov–Maxwell system

We consider continuation criteria for the three-dimensional relativistic Vlasov–Maxwell system. When the particle density, $f(t,x,p)$, is compactly supported at $t=0$, we prove ${6p_0^{\frac{18}{5r}?1+\beta }f6}_{L_t^{\infty }{L}_x^r{L}_p^1}?1$, where $1\le r\le 2$ and $\beta >0$ is arbitrarily small, is a continuation criteria. Our continuation criteria is an improvement in the $1\le r\le 2$ range to the previously best known criteria ${6p_0^{\frac{4}{r}?1+\beta }f6}_{L_t^{\infty }{L}_x^r{L^1}_p}?1$ due to Kunze [7]. We also consider continuation criteria when $f(0,x,p)$ has noncompact support. In this regime, Luk–Strain [9] proved that ${6p_0^{\theta }f6}_{L_x^1{L}_p^1}?1$ is a continuation criteria for $\theta >5$. We improve this result to $\theta >3$. Finally, we build on another result by Luk–Strain [8]. The authors proved boundedness of momentum support on a fixed two-dimensional plane is a sufficient continuation criteria. We prove the same result even if the plane varies continuously in time.     

On mixed Moore graphs

The Moore bound for a directed graph of maximum out-degree d and diameter k is $M_{d,k}=1+d+d^2+\cdots +d^k$. It is known that digraphs of order $M_{d,k}$ (Moore digraphs) do not exist for $d>1$ and $k>1$. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is $M_{d,k}^{*}=1+d+d(d-1)+\cdots +d(d-1{)}^{k-1}$. Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter $k=2$ are unique and that mixed Moore graphs of diameter $k⩾3$ do not exist.     

The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Chaos in rational systems in the plane containing quadratic terms

Consider the following system of rational equations containing quadratic termsChaos in the sense of Li–Yorke is considered. This is based on the Marotto’s theorem via obtaining a snap-back repeller. In fact, first in a special case when $F_1=F_2=0$, we show that origin is a snap-back repeller and so the system has chaotic behavior in the sense of Li–Yorke under some conditions. Then in a more general case, we prove that existence of chaos in the sense of Li–Yorke for the above system.