Ordinary fracture functions, describing hadrons production in the deep inelastic scattering target fragmentation region, are generalized to account for the production of hadrons in arbitrary number, thus offering a renewed framework for dealing with QCD initial state radiation. We also propose a new jet-like observable which measures beam remnants and low-$p_{perp}$ scattering fragments and derive its QCD evolution equations by using Jet Calculus. Possible implications for semi-inclusive deep inelastic scattering and hadron-hadron reactions are shortly discussed.
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