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The exact solution of a diffusion-reaction model for the trapping and annihilation of positrons in grain boundaries of polycrystalline materials with competitive trapping at intragranular point defects is presented. Closed-form expressions are obtained for the mean positron lifetime and for the intensities of the positron lifetime components associated with trapping at grain boundaries and at intragranular point defects. The closed-form solutions allow direct insight in the physical details of the positron annihilation characteristics and can be conveniently applied for the analysis of experimental data. It turns out that the model is not only essential for positron annihilation studies which aim at issues of grain-boundary physics or nanoscaled material but is also of relevance for studies of point defects in polycrystalline materials when grain sizes are in the micrometer range. more...