We explore the velocity fluctuations in a fluid due to a dilute suspension of randomly-distributed vortex rings at moderate Reynolds number, for instance those generated by a large colony of jellyfish. Unlike previous analysis of velocity fluctuations associated with gravitational sedimentation or suspensions of microswimmers, here the vortices have a finite lifetime and are constantly being produced. We find that the net velocity distribution is similar to that of a single vortex, except for the smallest velocities which involve contributions from many distant vortices; the result is a truncated 5/3-stable distribution with variance (and mean energy) linear in the vortex volume fraction φ. The distribution has an inner core with a width scaling as φ 3/5 , then long tails with power law |u| −8/3 , and finally a fixed cutoff (independent of φ) above which the probability density scales as |u| −5 , where u is a component of the velocity. We argue that this distribution is robust in the sense that the distribution of any velocity fluctuations caused by random forces localized in space and time has the same properties, except possibly for a different scaling after the cutoff.
Abstract. We provide conditions under which the set of Rijndael-like functions considered as permutations of the state space and based on operations of the finite field GF(p k ) (p ≥ 2) is not closed under functional composition. These conditions justify using a sequential multiple encryption to strengthen generalized Rijndael like ciphers. In [39], R. Sparr and R. Wernsdorf provided conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(2 k ) is equal to the alternating group on the state space. In this paper we provide conditions under which the group generated by the Rijndael-like round functions based on operations of the finite field GF(p k ) (p ≥ 2) is equal to the symmetric group or the alternating group on the state space.
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