Quantum relativistic decay law of moving unstable particle is analytically calculated in the model case of the BreitWigner mass distribution. It turns out that Einstein time dilation of the moving particle decay holds approximately at times when the decay is exponential. The related correction is calculated analytically. Being very small at these times it is practically unobservable. It is shown that Einstein dilation fails for large times t when decay is not exponential. An unstable system of the kind of K 0 meson (which is the superposition of K s and K l ) is also considered. In this case, the violation of Einstein dilation is shown to be appreciable at all times under some condition.The investigation has been performed at the Bogoliubov Laboratory of Theoretical Physics, JINR.

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Cauchy inequality (CI) relates scalar products of two vectors and their norms. I point out other similar inequalities (SI). Starting with CI Schroedinger (1930) derived his uncertainty relation (UR). By using SI other various UR can be obtained. It is shown that they follow from the Schroedinger UR. Two generalizations of CI are obtained for mixed states described by density matrices. Using them two generalizations of UR for mixed states are derived. Both differ from the UR generalization known in the literature.The discussion of these generalizations is given.

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