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Minwise hashing has become a standard tool to calculate signatures which allow direct estimation of Jaccard similarities. While very e cient algorithms already exist for the unweighted case, the calculation of signatures for weighted sets is still a time consuming task. BagMinHash is a new algorithm that can be orders of magnitude faster than current state of the art without any particular restrictions or assumptions on weights or data dimensionality. Applied to the special case of unweighted sets, it represents the rst e cient algorithm producing independent signature components. A series of tests nally veri es the new algorithm and also reveals limitations of other approaches published in the recent past.Sometimes it is more convenient to represent objects as bags also known as multisets, where each element is associated with a nonnegative weight. For example, words or shingles in text documents

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Micromagnetic simulations provide guidelines for the development of composite magnetic materials and magnetic devices. The granular structure of a material and the magnetic interactions between the different magnetic parts of a device can be easily taken into account by the finite element method and the boundary element method, respectively. The finite element equations are derived for the calculation of the magnetostatic field. Similarly, it is shown how the total energy of a magnetic system can be computed by matrix‐vector operations. Efficient methods for calculating equilibrium magnetic states and the simulation of the dynamic response of a magnetic material to a time varying external field are discussed. Numerical schemes for the simulation of magnetization dynamics in systems involving moving parts are introduced. The numerical methods are demonstrated by showing results of perpendicular magnetic recording simulation on composite media.