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.     

Self-similar solutions of stationary Navier–Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension $n=3,4$. For $n=3$, if the external force is axisymmetric, scaling invariant, $C^{1,\alpha }$ continuous away from the origin and small enough on the sphere $S^2$, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class $C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$. Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular ($U\in C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in $L^{4/3,\infty }({\mathbb{R}}^4)$.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].     

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.     

Refined weight of edges in normal plane maps

The weight $w\left(e\right)$ of an edge $e$ in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge $e=uv$ is of type $(i,j)$ if $d\left(u\right)\le i$ and $d\left(v\right)\le j$. In 1940, Lebesgue proved that every NPM has an edge of one of the types $(3,11)$, $(4,7)$, or $(5,6)$, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge $e$ with $w\left(e\right)\le 13$, which bound is sharp. Borodin (1989), answering Erd?s’ question, proved that every NPM has either a $(3,10)$-edge, or $(4,7)$-edge, or $(5,6)$-edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge $e$ with $w\left(e\right)\le 12$. Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types $(3,9)$, $(4,7)$, or $(5,6)$.By a $k^{\left(?\right)}$-vertex we mean a $k$-vertex incident with precisely $?$ triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: $(3^{\left(3\right)},10)$, $(3^{\left(2\right)},9)$, $(3^{\left(1\right)},7)$, $(4^{\left(4\right)},7)$, $(4^{\left(3\right)},6)$, $(5^{\left(5\right)},6)$, or $(5,5)$, where all bounds are best possible. In particular, this implies that the bounds in $(3,10)$, $(4,7)$, and $(5,6)$ can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively.