For the purpose of calculating (ro-)vibrational spectra, rate constants, scattering cross sections, etc. product basis sets are very popular. They, however, have the important disadvantage that they are unusably large for systems with more than four atoms. In this paper we demonstrate that it is possible to efficiently use a basis set obtained by removing, from a product basis set, functions associated with the largest diagonal Hamiltonian matrix elements. This is done by exploiting the fact that for every factor of every term in the Hamiltonian, there is a basis-set order in which the matrix representation of the factor is block diagonal. Due to this block diagonality the Lanczos algorithm can be implemented efficiently. Tests with model Hamiltonians with as many as 32 coordinates illustrate the merit of the ideas.
In this paper we develop a leader election protocol P with the following features:1. The protocol runs in the perfect information model: Every step taken by a player is visible to all others.2. It has linear immunity: If P is run by n players and a coalition of cln players deviates from the protocol, attempting to have one of them elected, their probability of success is < 1-c2, where c 1 , c 2 ~ 0 are absolute constants.3. It is fast: The running time of P is polylogarithmic in n, the number of players.A previous protocol by Alon and Naor achieving linear immunity in the perfect information model has a linear time complexity. The main ingredient of our protocol is a reduction subprotocol. This is a way for n players to elect a subset of themselves which has the following property. Assume that up to en of the players are bad and try to have as many of them elected to the subset. Then with high probability, the fraction of bad players among the elected ones will not exceed e in a significant way. The existence of such a reduction protocol is first established by a probabilistic argument. Later an explicit construction is provided which is based on the spectral properties of Ramanujan graphs.
MYRG2015-00203-FED). The opinions expressed herein are those of the authors.The first two authors wrote the chapter with the collaboration of other working group members.
Historical BackgroundZou (2015) summarised findings from historical investigations of arithmetic in ancient China, including how number units were derived and named and how numbers were represented with rod or bead calculation tools and with symbols. Siu (2015) studied the book of Tongwen Suanzhi (同文算指) (Rules of Arithmetic Common to Cultures, 1614) and reviewed how counting rods and the abacus were gradually replaced with written calculations in China. Sun (2015), also discussing early Chinese development, presented the use of advanced number names and
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