In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq 540t^{2}+\frac{175811}{2}t+\frac{7989}{2}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2 \over 5}},$ which is better than the previous bounds $\sqrt 2$ (see [2]), $\sqrt {2+{2562\over 6911}}$ (see [7]).

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The alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex. It converges quickly to an approximate solution, but can take a long time to converge to a solution of high‐accuracy. Previously, Anderson acceleration has been applied to ADMM, by treating it as a fixed‐point iteration for the concatenation of the dual variables and a subset of the primal variables. In this paper, we note that the equivalence between ADMM and Douglas‐Rachford splitting reveals that ADMM is in fact a fixed‐point iteration in a lower‐dimensional space. By applying Anderson acceleration to such lower‐dimensional fixed‐point iteration, we obtain a more effective approach for accelerating ADMM. We analyze the convergence of the proposed acceleration method on nonconvex problems, and verify its effectiveness on a variety of computer graphics including geometry processing and physical simulation.

In the description of isolated gravitating systems in General Relativity a spacelike timeslice has the structure of a complete Riemannian three-manifold with an asymptotically at end. Let (N; g) denote an end of such a complete Riemannian three-manifold, ie N is diffeomorphic to R3\B1 (0), and the metric on N asymptotically approaches the Euclidean metric near infinity: gij=