Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O\left(k^5\right)}?n^9$. In this article we improve this running time to $2^{O\left(k^2\right)}?n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O\left(k^5\right)$ to $O\left(k^2\right)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p?1$ for $p\ge 3$, we speed up the running time to $2^{O(k?k^{\frac{1}{p?1}})}?n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p\ge 3$, Weighted Independent Set cannot be solved in time $2^{o\left(k\right)}?n^{O\left(1\right)}$ in the class of $\{\text{bull},C_4,\dots ,C_{2p?1}\}$-free graphs unless the ETH fails.     

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0,y^0\in K$, compute sequences $\left\{x^k\right\}$ and $\left\{y^k\right\}$ such that $x^{k+1}=(1-a^k)x^k+a^k{P}_K[y^k-\rho T\left(y^k\right)]\text{for }\rho >0$$y^k=(1-b^k)x^k+b^k{P}_K[x^k-\eta T\left(x^k\right)]\text{for }\eta >0\text{,}$ where $T:K\to H$ is a nonlinear mapping on $K,P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$. The two-step model is applied to some variational inequality problems.     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$     

Minimal degree univariate piecewise polynomials with prescribed Sobolev regularity

For $k\in \{1,2,3,\dots \}$, we construct an even compactly supported piecewise polynomial ${\psi }_k$ whose Fourier transform satisfies $A_k{(1+{\omega }^2)}^{?k}\le {\stackrel{?}{\psi }}_k\left(\omega \right)\le B_k{(1+{\omega }^2)}^{?k}$, $\omega \in \mathbb{R}$, for some constants $B_k\ge A_k>0$. The degree of ${\psi }_k$ is shown to be minimal, and is strictly less than that of Wendland’s function ${?}_{1,k?1}$ when $k>2$. This shows that, for $k>2$, Wendland’s piecewise polynomial ${?}_{1,k?1}$ is not of minimal degree if one places no restrictions on the number of pieces.     

A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

The NLC-width and clique-width for powers of graphs of bounded tree-width

The $k$-power graph of a graph $G$ is a graph with the same vertex set as $G$, in that two vertices are adjacent if and only if, there is a path between them in $G$ of length at most $k$. A $k$-tree-power graph is the $k$-power graph of a tree, a $k$-leaf-power graph is the subgraph of some $k$-tree-power graph induced by the leaves of the tree.We show that (1) every $k$-tree-power graph has NLC-width at most $k+2$ and clique-width at most $k+2+max\{?\frac{k}{2}??1,0\}$, (2) every $k$-leaf-power graph has NLC-width at most $k$ and clique-width at most $k+max\{?\frac{k}{2}??2,0\}$, and (3) every $k$-power graph of a graph of tree-width $l$ has NLC-width at most ${(k+1)}^{l+1}?1$, and clique-width at most $2?{(k+1)}^{l+1}?2$.