This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c : X →

Abstract. We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed. This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the hyperfinite ones. These are by definition the increasing unions of sequences of Borel equivalence relations with finite equivalence classes but equivalently they can be also described as the ones induced by the orbits of a single Borel automorphism.

An analytical model for the impulse of a single-cycle pulse detonation tube has been developed and validated against experimental data. The model is based on the pressure history at the thrust surface of the detonation tube. The pressure history is modeled by a constant pressure portion, followed by a decay due to gas expansion out of the tube. The duration and amplitude of the constant pressure portion is determined by analyzing the gasdynamics of the self-similar ow behind a steadily moving detonation wave within the tube. The gas expansion process is modeled using dimensional analysis and empirical observations. The model predictions are validated against direct experimental measurements in terms of impulse per unit volume, speci c impulse, and thrust. Comparisons are given with estimates of the speci c impulse based on numerical simulations. Impulse per unit volume and speci c impulse calculations are carried out for a wide range of fuel-oxygen-nitrogen mixtures (including aviation fuels) of varying initial pressure, equivalence ratio, and nitrogen dilution. The effect of the initial temperature is also investigated. The trends observed are explained using a simple scaling analysis showing the dependency of the impulse on initial conditions and energy release in the mixture. = time taken by the rst re ected characteristic to reach the thrust surface t 3 = time associated with pressure decay period t ¤ = time at which the rst re ected characteristic exits the Taylor wave U CJ = Chapman-Jouguet detonation velocity u = ow velocity u e = exhaust velocity u 2 = ow velocity just behind detonation wave V = volume of gas within detonation tube X F = fuel mass fraction ® = nondimensional parameter corresponding to time t 2 = nondimensional parameter corresponding to pressure decay period°= ratio of speci c heats 1P = pressure differential 1P 3 = pressure differential at the thrust surfacé = similarity variablȩ = cell size 5 = nondimensional pressure ½ e = exhaust density ½ 1 = initial density of reactants ¿ = nondimensional time ct=L Á = equivalence ratio

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= pendulum mass P = pressure P env = environment pressure P lip = pressure on lip at exit of tube P TS = pressure on thrust surface in detonation tube interior P 1 = initial pressure of reactants P 2 = Chapman-Jouguet pressure P 3 = pressure of burned gases behind Taylor wave p = pitch of spiral obstacles S = wetted surface area of tube's inner diameter T 1 = initial temperature of reactants t = time U CJ = Chapman-Jouguet detonation velocity V = internal volume of detonation tubē = ratio of N 2 -to-O 2 concentration in initial mixture°= ratio of speci c heats in combustion products 1x = horizontal pendulum displacemenţ = cell size ½ 1 = density of combustible mixture at the initial temperature and pressure ¿ = wall shear stress