The product ring Q S = p∈S Q p , where S is a finite set of prime numbers and Q p is the field of p-adic numbers, is a second countable, locally compact and totally disconnected topological ring. This work introduces a natural ultrametric on Q S that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2 (Q S ). The fundamental solution of this equation is a normal transition function of a Markov process on Q S . As a result, the techniques developed here provides a general framework of these problems on other related ultrametric groups.

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In this paper we continue the study of evolutoids of convex curves. We proved that if a curve is homothetic to one of its evolutoids then it is a circle. This result is analogous, for the case of evolutoids, to the planar case of the famous homothetic floating body problem which states that if a floating body is homothetic to the body itself then it is an ellipsoid. Among other things, we proved that a curve and any of its evolutoids have the same Steiner point. Moreover, some relations between evolutoids and constant angle caustics are also given, for instance, that if for a given angle the left and right evolutoids describe the same curve then the curve possesses a constant angle caustic.Mathematics Subject Classification. 53A04.

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