In this paper, we consider algorithms to pack rectangles into a strip. As the main result we present an algorithm that packs rectangles online and for which the ratio of expected wasted area to expected occupied area tends to zero as the number of rectangles increases. The research was supported by the Russian Foundation for Basic Research, grants 05–01–00798 and 04–01–00359. [ABSTRACT FROM AUTHOR]
In this paper we present two recent results for matrices arising in the mixed finite element method on triangular meshes. The first result is an algebraic proof for the discrete LBB condition. The second one is a new algorithm for the construction of a two-level spectrally equivalent preconditioner for the condensed matrix. [ABSTRACT FROM AUTHOR]
The distribution of jobs in a system with m identical parallel processors which minimises the load of the maximally loaded processor is an NP-hard problem. Many approximate algorithms are developed for this problem, but for the version of the problem where the jobs arrive ad must be treated on-line there is no algorithm possessing the guaranteed estimate which is less than 1 + 1 / ... for m ≥ 4 and tends to 1.837 as m → ∞. In this paper, we consider the version of the problem where jobs arrive one by one and must be treated on-line under the additional condition that the total duration of the jobs is known. For this version of the problem we suggest an algorithm with the guaranteed estimate equal to 5/3. [ABSTRACT FROM AUTHOR]