A generalization of the original Diffie-Hellman key exchange in (Z/pZ) * found a new depth when Miller [27] and Koblitz [16] suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general.In Section 5 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.

In this paper, we compute the natural density of the set of k × n integer matrices that can be extended to an invertible n × n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form O k×(n−k) I k . Connections with Cesàro's Theorem on the density of coprime integers and Quillen-Suslin's Theorem are also presented.

Acoustic scattering from an isotropic elastic hollow cylindrical shell of infinite length excited by an obliquely incident plane acoustic wave is investigated. The form functions of an aluminum c3/lindrical shell immersed in water have been calculated by the direct summation of the Rayleigh series. Computations are made at angles (with the normal to the cylinder axis) between a = 0 ø and a = 35 ø. The results of the theoretical calculation are in good agreement with the results of experiments. The experimental results have shown in a frequency range of kl a = 0-20 that the resonances are related to three wave families: the circumferential wave (l = 2) detected for angles smaller than the "angle of longitudinal wave in thin rods" (a•), the guided wave (p = 1 ) detected for angles smaller than the second critical angle (at), and the Scholte-Stoneley wave (l = 0). The evolution of the resonance frequencies is followed for different angles and one can note experimentally, that at an angle superior to the Rayleigh critical angle (a = 30.3ø), resonances of the Scholte-Stoneley wave have been observed.

Using a pulse reflection technique an ultrasonic system has been developed to monitor in situ the coagulation process of rennetted milk. The velocity and attenuation of ultrasonic waves through coagulating milk were continuously monitored. The observed changes in ultrasonic velocity during coagulation were used to predict the coagulation time. The coagulation time is indicative of the transition from the enzymatic phase to the physicochemical phase. The determination of coagulation time has a decisive role in determining the qualities of the end product in cheesemaking.