The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when $k$ is odd, it happens that the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a $k$-uniform connected hypergraph are given. For a connected $k$-uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian H-eigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if $k$ is even and the hypergraph is odd-bipartite.Comment: 26 pages, 3 figure

Surjective endomorphisms of projective surfaces -- the existence of infinitely many dense orbits

Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$. Using this, we extend the second author's result to singular surfaces to the extent that either $X$ has an $f$-invariant non-constant rational function, or $f$ has infinitely many Zariski-dense forward orbits; this result is also extended to Adelic topology (which is finer than Zariski topology)

Surface plasmon polaritons in topological insulator

We study surface plasmon polaritons on topological insulator-vacuum interface. When the time-reversal symmetry is broken due to ferromagnetic coupling, the surface states exhibit magneto-optical Kerr effect. This effect gives rise to a novel transverse type surface plasmon polariton, besides the longitudinal type. In specific, these two types contain three different channels, corresponding to the pole of determinant of Fresnel reflection matrix. All three channels of surface plasmon polaritons display tight confinement, long lifetime and show strong light-matter coupling with a dipole emitter.Comment: 6 pages, 4 figure