By David M. Burton

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**Extra resources for An introduction to abstract mathematical systems**

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V. Jagadish (1988): Partitioning techniques for largegrained parallelism. IEEE Trans. Computers 37, 1627–1634. [4] W. N. K. L. K. Sitaraman (2001): Augmented ring networks. IEEE Trans. Parallel and Distr. Systs. 12, 598–609. [5] S. Akl (1989): The Design and Analysis of Parallel Algorithms. Prentice-Hall, Englewood Cliffs, NJ. [6] R. L. Rosenberg (1982): On embedding rectangular grids in square grids. IEEE Trans. , C-31, 907–913. [7] A. I. E. Schauser, C. Scheiman (1997): LogGP: incorporating long messages into the LogP model for parallel computation.

A. Case study [132]: This source studies an unusual facet of the security problem in WC. It develops a computationally lightweight scheme for keeping track of which volunteers in a WC project computed which tasks. Much of the scheme employs familiar algorithmic techniques involving search trees for pointand range-queries. The unique aspect of the scheme is a strategy that assigns positive-integer indices to 1. the set of all tasks at the master site, 2. all volunteers (who are allowed to arrive and depart dynamically), and 3.

The timeline (not to scale) for 3 “rented” workstations, indicating each workstation’s lifespan. Note that each Pi’s lifespan is partitioned in the figure between its incarnations as some Ps and some Pf . a b a computationally tractable, perspicuous way. 3, the work production of the protocol P (R, U) that is specified by the startup indexing Σ = 〈 s1, s2, . , s n 〉 and finishing indexing Φ = 〈 f1, f2, . , fn 〉 over a lifespan of duration L is given by the following system of linear equations: JVC + t B1, 2 1 K 1 VC2 + t2 K B2, 1 K h h K B B K n - 1, 1 n - 1, 2 K Bn, 1 Bn, 2 L where g B1, n NO JK w1 NO JK L - (c1 + 2) FC NO g B2, n O K w2 O K L - (c2 + 2) FC O O$ K h O= K O, g h h O K O K O g Bn - 1, n O K wn - 1O K L - (cn - 1 + 2) FCO g VCn + tnO K wn O K L - (cn + 2) FC O P L P L P ● SBi is the set of startup indices of workstations that start before Pi ; ● FAi is the set of finishing indices of workstations that finish after Pi ; ● ● def SBi + FAi ; and Z ] r0 + x + xd if j !

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