The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.     

On the dominating set polytope of web graphs

The dominating set polyhedron of a web graph $W_n^k$ is the set covering polyhedron of a circulant matrix $C_n^{2k+1}$. Working on the set covering polyhedron of general circulant matrices, Argiroffo and Bianchi found a class of valid inequalities, induced by a particular family of circulant minors. The authors also identified sufficient conditions on the minors in order to induce facet defining inequalities. In this work we generalize these results for new valid inequalities associated with every circulant minor. We conjecture that, for any k, the minor inequalities together with the boolean facets are enough to describe the set covering polyhedron of $C_n^k$. This conjecture is true for $k=3$ as Bouchakour et al. prove working in the context of the Minimum Weight Dominating Set problem (MWDSP) on cycles, i.e, on webs $W_n^1$. They also derive the polynomiality of MWDSP on cycles by giving a polynomial separation algorithm for minor inequalities of $C_n^3$. Although the computational complexity of the separation problem of every minor inequality is still an open problem, we can state that the set covering problem on $C_n^k$ matrices is polynomial for a fixed k and obtain polynomial separation algorithms for particular classes of minor inequalities.     

A cohomological study of local rings of embedding codepth 3

The generating series of the Bass numbers ${\mu }_R^i={\mathrm{rank}}_k{\mathrm{Ext}}_R^i(k,R)$ of local rings $R$ with residue field $k$ are computed in closed rational form, in case the embedding dimension $e$ of $R$ and its depth $d$ satisfy $e?d\le 3$. For each such $R$ it is proved that there is a real number $\gamma >1$, such that ${\mu }_R^{d+i}\ge \gamma {\mu }_R^{d+i?1}$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where ${\mu }_R^{d+2}={\mu }_R^{d+1}=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.     

Monochromatic loose path partitions in k-uniform hypergraphs

A conjecture of Gyárfás and Sárközy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k?2$ vertices. A weaker form of this conjecture with $2k?5$ uncovered vertices instead of $k?2$ is proved. Thus the conjecture holds for $k=3$. The main result of this paper states that the conjecture is true for all $k\ge 3$.     

On 3-stable number conditions in n-connected claw-free graphs

For a subgraph $X$ of $G$, let ${\alpha }_G^3\left(X\right)$ be the maximum number of vertices of $X$ that are pairwise distance at least three in $G$. In this paper, we prove three theorems. Let $n$ be a positive integer, and let $H$ be a subgraph of an $n$-connected claw-free graph $G$. We prove that if $n\ge 2$, then either $H$ can be covered by a cycle in $G$, or there exists a cycle $C$ in $G$ such that ${\alpha }_G^3(H?V\left(C\right))\le {\alpha }_G^3\left(H\right)?n$. This result generalizes the result of Broersma and Lu that $G$ has a cycle covering all the vertices of $H$ if ${\alpha }_G^3\left(H\right)\le n$. We also prove that if $n\ge 1$, then either $H$ can be covered by a path in $G$, or there exists a path $P$ in $G$ such that ${\alpha }_G^3(H?V\left(P\right))\le {\alpha }_G^3\left(H\right)?n?1$. By using the second result, we prove the third result. For a tree $T$, a vertex of $T$ with degree one is called a leaf of $T$. For an integer $k\ge 2$, a tree which has at most $k$ leaves is called a $k$-ended tree. We prove that if ${\alpha }_G^3\left(H\right)\le n+k?1$, then $G$ has a $k$-ended tree covering all the vertices of $H$. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).     

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.     

Collision and escape orbits in a generalized Hénon–Heiles model

The motion of a material point of unit mass in a field determined by a generalized Hénon–Heiles potential $U=A{q}_1^2+B{q}_2^2+C{q}_1^2{q}_2+D{q}_2^3$, with $(q_1,q_2)=$ standard Cartesian coordinates and $(A,B,C,D)\in {(0,\infty )}^2\times R^2$, is addressed for two limit situations: collision and escape. Using McGehee-type transformations, the corresponding collision and infinity boundary manifolds pasted on the phase space are determined. These are fictitious manifolds, but, due to the continuity with respect to initial data, their flow determines the near by orbit behaviour.The dynamics on the collision and infinity manifolds is fully described. The topology of the flow on the collision manifold is independent of the parameters. In the full phase space, while spiraling collision orbits are present, most of the orbits avoid collision. The topology of the flow on the infinity manifold changes as the ratio between $C$ and $D$ varies. More precisely, there are two symmetric pitchfork bifurcations along the line $2C?3D=0$, due to the reshaping of the potential along the bifurcation line. Besides rectilinear and spiraling orbits, the near-escape dynamics includes oscillatory orbits, for which angular momentum alternates sign.