We study the tropical lines contained in smooth tropical surfaces in ℝ^{3}. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however, there exist smooth tropical surfaces of any degree with infinitely many tropical lines.

No abstract uploaded!

nthispaperwewillpresentthelocalstabilityanalysisandlocalerrorestimatefor the local discontinuous Galerkin (LDG) method, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Based on a general framework on the local stability, we obtain the optimal error estimate out of the local subdomain, which is nearby the outflow boundary point and has the width of O(h log(1/h)), for the semi-discrete LDG scheme and the fully-discrete LDG scheme with the second order explicit Runge–Kutta time-marching. Here h is the maximum mesh length. The numerical experiments are given also.

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to $-1$, independently of the genus. This provides a counterexample to some suggestions that the Weil-Petersson metric becomes asymptotically flat, as the genus $g$ goes to infinity, in the thick loci in \tec space. Adopting a different perspective on curvature, we also show that the minimal (most negative) eigenvalue of the curvature operator at any point in the \tec space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius.

In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of \mathbb {R} P^\infty with the aid of a computer, which gives us its Adams \mathbb {R} P^\infty -page in the range of \mathbb {R} P^\infty . We also compute the transfer map on the Adams \mathbb {R} P^\infty -pages. These data are used in our computations of the stable homotopy groups of \mathbb {R} P^\infty in [6] and of the stable homotopy groups of spheres in [7].