On two Diophantine inequalities over primes

Let $1<c<37∕18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N]$, the Diophantine inequality $|p_1^c+p_2^c+p_3^c?R|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3$. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c?N|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.     

Further results on logarithmic terraces

Let p be an odd prime, and let x be a primitive root of p. Suppose that we write the elements of ${\mathbb{Z}}_{p-1}$ as $1,2,\dots ,p-1,$ and that, wherever we evaluate $x^l(\mathrm{mod}p)$, we always write it as one of $1,2,\dots ,p-1.$ Let $?=(l_1,\dots ,l_{p-1})$ be a terrace for ${\mathbb{Z}}_{p-1}$. Then $?$ is said to be a logarithmic terrace if $\mathbf{e}=(e_1,\dots ,e_{p-1})$, defined by $e_i\equiv x^{l_i}(\mathrm{mod}p)$, is also a terrace for ${\mathbb{Z}}_{p-1}$. We study properties of logarithmic terraces, in particular investigating terraces which are simultaneously logarithmic for two different primitive roots.     

Cycles with a chord in dense graphs

A cycle of order $k$ is called a $k$-cycle. A non-induced cycle is called a chorded cycle. Let $n$ be an integer with $n\ge 4$. Then a graph $G$ of order $n$ is chorded pancyclic if $G$ contains a chorded $k$-cycle for every integer $k$ with $4\le k\le n$. Cream, Gould and Hirohata (Australas. J. Combin. 67 (2017), 463–469) proved that a graph of order $n$ satisfying ${deg}_{G}u+{deg}_{G}v\ge n$ for every pair of nonadjacent vertices $u$,  $v$ in $G$ is chorded pancyclic unless $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, the Cartesian product of $K_3$ and $K_2$. They also conjectured that if $G$ is Hamiltonian, we can replace the degree sum condition with the weaker density condition $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ and still guarantee the same conclusion. In this paper, we prove this conjecture by showing that if a graph $G$ of order $n$ with $\left|E\left(G\right)\right|\ge \frac{1}{4}{n}^2$ contains a $k$-cycle, then $G$ contains a chorded $k$-cycle, unless $k=4$ and $G$ is either $K_{\frac{n}{2},\frac{n}{2}}$ or $K_3\square K_2$, Then observing that $K_{\frac{n}{2},\frac{n}{2}}$ and $K_3\square K_2$ are exceptions only for $k=4$, we further relax the density condition for sufficiently large $k$.     

A scheme for interpolation by Hankel translates of a basis function

This work discusses interpolation of complex-valued functions defined on the positive real axis $I$ by certain special subspaces, in a variational setting that follows the approach of Light and Wayne [W. Light, H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates, Numer. Math. 81 (1999) 415–450]. The set of interpolation points will be a subset $\{a_1,\dots ,a_n\}$ of $I$ and the interpolants will take the form $u\left(x\right)=\sum _{i=1}^n{\alpha }_i\left({\tau }_{a_i}?\right)\left(x\right)+\sum _{j=0}^{m?1}{\beta }_j{p}_{\mu ,j}\left(x\right)(x\in I)\text{,}$ where $\mu \ge ?1/2,?$ is a complex function defined on $I$ (the so-called basis function), $p_{\mu ,j}\left(x\right)=x^{2j+\mu +1/2}(j\in {\mathbb{Z}}_{+},0\le j\le m?1)$ is a Müntz monomial, ${\tau }_z(z\in I)$ denotes the Hankel translation operator of order $\mu$, and ${\alpha }_i,{\beta }_j(i,j\in {\mathbb{Z}}_{+},1\le i\le n,0\le j\le m?1)$ are complex coefficients. An estimate for the pointwise error of these interpolants is given. Some numerical examples are included.     

The multipartite Ramsey number for the 3-path of length three

We study the Ramsey number for the 3-uniform loose path of length three, $P_3^3$, and $n$ colors. We show that $R(P_3^3;n)\le {\lambda }_{0}n+7\sqrt{n}$, for some explicit constant ${\lambda }_0=1.97466\dots$.