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Given a real number α ∈ (0 , 1) , we define the Webster sequence of density α to be W α = (⌈ (n − 1 / 2) / α ⌉) n ∈ N , where ⌈ x ⌉ is the ceiling function. It is known that if α and β are irrational with α + β = 1 , then W α and W β partition N. On the other hand, an analogous result for three-part partitions does not hold: There does not exist a partition of N into sequences W α , W β , W γ with α , β , γ irrational. In this paper, we consider the question of how close one can come to a three-part partition of N into Webster sequences with prescribed densities α , β , γ. We show that if α , β , γ are irrational with α + β + γ = 1 , there exists a partition of N into sequences of densities α , β , γ , in which one of the sequences is a Webster sequence and the other two are "almost" Webster sequences that are obtained from Webster sequences by perturbing some elements by at most 1. We also provide two efficient algorithms to construct such partitions. The first algorithm outputs the n th element of each sequence in O (1) time and the second algorithm gives the assignment of the m th positive integer to a sequence in O (1) time. We show that the results are best-possible in several respects. Moreover, we describe applications of these results to apportionment and optimal scheduling problems. [ABSTRACT FROM AUTHOR]

• A fast contact detection algorithm (FCDA) for ice-ship interaction (ISI) is proposed • The MPI parallelization for 2-D Peridynamic (PD) theory is first introduced • The newly developed ice-ship interaction (ISI) model is effective in predicting ice damage and crack propagation in the icebreaking process • The developed ice-ship interaction (ISI) model is applied to investigate icebreaking effectiveness for two different bow shapes The bow shape is the most critical factor to determine the icebreaking performance of an icebreaker. Mechanism study on the icebreaking process for different bow types is necessary for the initial design of the icebreaker hull form. This paper proposed an ice-ship interaction model based on the meshfree method, Peridynamics, in which the geometric mathematics concept is embedded to detect the contact between material points and ship hull. Furthermore, a fast contact detection algorithm based on Massage Passing Interface (MPI) solver is built to improve the computational efficiency of the developed numerical method. Two typical icebreaker bows, the conventional bow and the unconventional bow, breaking the level ice with constant speed is numerically studied by the above model. The results of the conventional icebreaker bow are compared with the experimental results, which verifies the simulation accuracy of the model developed in the present work. Afterwards, the icebreaking modes and icebreaking loads of two different shapes of icebreaker bows are compared and analysed. The results show that the developed ice-ship interaction model effectively predicts differences of icebreaking processes between different icebreakers, such as ice damage pattern, ice loads, and channel, despite their common point in domain bending failure mode. Moreover, this research significantly improves computational efficiency and provides theoretical guidance for designing the icebreaker bow. [ABSTRACT FROM AUTHOR]