The scaling and relaxation behavior around the fishtail minimum is studied in detail in a wide temperature range ͑3-70 K͒ on DyBa 2 Cu 3 O 7Ϫ␦ single crystals exhibiting a pronounced fishtail effect. Magnetic hysteresis loops ͑MHL's͒ normalized with respect to the height and position of the fishtail maximum fall on a universal curve which form can be derived from the phenomenological model of a thermally activated flux creep proposed by Perkins et al. ͓Phys. Rev. B 51, 8513 ͑1995͔͒. This universal curve tends at low fields towards zero. At low temperatures, the drop of j s at low fields is usually masked by a wide central peak. By subtracting the universal curve from the experimental j s (B) data we separate the contribution of the central peak. It has a simple, exponentially decaying field dependence. This implies that the fishtail minimum at low fields might be understood as a result of an overlapping of two contributions originating from separate pinning mechanisms: one active mainly at high fields and dying away with B going to zero and another one ͑responsible for the central peak of the MHL͒ vanishing rapidly with increasing field. This concept is also supported by relaxation experiments. These experiments confirm that the shape of MHL's is given by a dynamic equilibrium between the induction, pinning, and relaxation processes. ͓S0163-1829͑97͒05305-8͔

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Exact analytical results are obtained for the magnetization of a superconducting thin strip with a general behavior Jc(B) of the critical current density. We show that within the critical-state model the magnetization as function of applied field, Ba, has an extremum located exactly at Ba = 0. This result is in excellent agreement with presented experimental data for a YBa2Cu3O 7−δ thin film. After introducing granularity by patterning the film, the central peak becomes shifted to positive fields, Bcp > 0, on the descending field branch of the loop. Our results show that a positive Bcp is a definite signature of granularity in superconductors.