The k Most Frequent Distances in the Plane

   Abstract. A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is f _n (k)=O(n ^1.4571 k ^.6286 ) improving on an earlier bound of Akutsu, Tamaki, and Tokuyama.     

The Most Critical Path in a PERT Network

The classical PERT approach uses the path with the largest expected duration as the critical path to estimate the probability of completing a project by a given deadline. However, in general, such a path is not the ‘most critical’ path and does not provide the smallest estimate for the probability of completion time. This paper studies the ‘most critical path’ problem and formulates it as an optimal path problem in a deterministic network with a two-attribute fractional objective function. An exact solution approach is presented for the optimal path problem which also gives the solution to the most critical path problem. The illustrative examples as well as our computational results demonstrate that the proposed algorithm provides estimates for the probabilities of completion time that are much more accurate than those of the classical approach.     

The k Most Frequent Distances in the Plane

Abstract. A new upper bound is shown for the number of incidences between n points and n families of concentric circles in the plane. As a consequence, it is shown that the number of the k most frequent distances among n points in the plane is f _n (k)=O(n ^1.4571 k ^.6286 ) improving on an earlier bound of Akutsu, Tamaki, and Tokuyama.     

数学建模竞赛论文的写作方法

本文针对数学建模竞赛论文的评阅要求和近年来参赛论文的实际情况,首先简要介绍了科技论文的写作规范,然后详细说明了竞赛论文的写作方法和注意事项,可供拟参加数学建模竞赛的学生和指导老师参考。     

Most robust M-estimators in the infinitesimal sense

Summary The change-of-variance curve (CVC) is generalized to M-estimators with piecewise continuous -functions, in which case it becomes a Schwartz distribution. An M-estimator is called most B-robust when it minimizes Hampel's gross-error sensitivity ^*, and most V-robust when it minimizes the change-of-variance sensitivity k ^*. In the general case, the median is most B-robust and most V-robust. If consideration is restricted to redescending M-estimators, then the skipped median is most B-robust and the median-type tanh-estimator is most V-robust. By means of these results, complete solutions of the problems of optimal infinitesimal robustness are obtained.     

Most vital links and nodes in weighted networks

The n most vital links (or nodes) in a weighted network are those n links (nodes) whose removal from the network results in the greatest increase in shortest distance between two specified nodes. Preliminary results are presented for obtaining these entities.     

The Average Condition Number of Most Tensor Rank Decomposition Problems is Infinite

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks \(r\ge 3\) as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions.