The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge-Kutta time discretization up to third order accuracy, for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent of the mesh-size $h$, even though the advection term is treated explicitly. The last is that under this time step condition, we obtain optimal error estimates in both space and time for the third order IMEX Runge-Kutta time-marching coupled with LDG spatial discretization. Numerical experiments are also given to verify the main results.

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We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |\chi|^{2}$. In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is $2|\chi|(|\chi|+1)$.

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We prove the following conjecture of Furstenberg (1969): if 𝐴,𝐵⊂[0,1] are closed and invariant under ×𝑝 mod1 and ×𝑞 mod1, respectively, and if log𝑝/log𝑞∉ℚ, then for all real numbers 𝑢 and 𝑣, dim_H (𝑢𝐴 + 𝑣) ∩ 𝐵 ≤ max {0, dim_H 𝐴 + dim_H 𝐵 − 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on ℝ. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.