Two-step Iteration Research Articles

Two consequences of determinacy consistent with choice

We begin with a ground model satisfying ZF + AD + ACR, and from it construct a generic extension satisfying ZFC + 6? = C2 + nonstationary ideal on coI is Co2-saturated. 0. Introduction. The of determinacy (henceforth AD) implies many interesting propositions about small cardinal numbers, for example c and (O2. It is natural to ask whether these propositions are true or, more realistically, can be true in the full universe V of sets. Our working hypothesis is that in V the axiom of choice is true and all games are determined, so the question becomes whether these propositions are consistent with ZFC together with determinacy. (We shall be vague about definable when we can afford to.) In this paper we shall consider two such consequences of AD, the first being that 2I = W2 and the second that the nonstationary ideal on xl is o2-saturated. We shall show that the theory consisting of ZFC + ADL(R) together with these two propositions is consistent. (ADL(R) is the assertion that all games in L(R), a large class of games, are deterniined.) In consonance with our working hypothesis, we would like to prove our result assuming only the consistency of ZFC + ADL(R), or equivalently, that of ZF + AD + DC (cf. ?1). At present we cannot do this. We instead assume the consistency of ZF + AD + ACR' where ACR is the axiom of choice for families indexed by the reals. Consistency-wise, ZF + AD + ACR is one of the strongest theories known to man; we show in ?1 that it proves the consistency of ZF + AD + DC. The only upper bound we know on the consistency strength of ZF + AD + ACR is that of ZF + ADR + 0 is regular. We offer some partial justification for our use of such a strong hypothesis at the end of the paper. Our consistency proof is by forcing. We start with a ground model M satisfying ZF + AC + ACR. By a two-step iteration we construct a generic extension M[G] onto so that 3h (h: 2 -RM). The forcing will add no new reals and preserve Co2, i.e., RM = RM[G] and (M =1M[G]. We then pass to the inner model N = L(RM, h) and show it has the desired properties. The paper is organized as follows: In ?1 we give some preliminaries and background information. In particular, we set forth those consequences of AD whose Received by the editors February 18, 1980. 1980 Mathematics Subject Classification. Primary 03E35; Secondary 03E60. ?1 982 American Mathematical Society 0002-9947/8 1 /001000/$05.75

Schlieren-Cinematographic and Holographic Diagnostic of a Laser-Produced Plasma in Xenon

We have studied extensively a gas breakdown in xenon produced by a giant pulse ruby laser with a power <100 MW. Detailed information concerning the structure of the laser plasma formation and of the following expansion has been obtained by different optical methods, including schlieren techniques (single frames and ultra high speed cinematography) and holography. The high quality of the holographic pictures was achieved by the use of a monomode laser. With this we were able to visualize the plasma history and to determine the velocities of the boundary layer and of the blast wave with utmost accuracy. Investigations of the electron density and electron temperature showed that a relaxation time of about 10 nsec is necessary to establish local thermodynamic equilibrium states. After this relaxation time it is then possible to carry out thermodynamic calculations, applying the shock-wave theory, to relate the optically measured expansion velocity with the plasma parameters involved. The mean specific internal energy epsilon , for instance, attained values in excess of 10(12) erg/g which decayed rapidly during the first 100 nsec to about 5. 10(10) erg/g. By comparing the results to theoretical calculations of we obtained a first estimate of the temperature, taking into consideration the partial densities n(j) as well as the partition functions Z(j)((i)) of the xenon atoms, the single-charged ions, and the double-charged ones. Furthermore, a two-step iteration computer program was used to give more detailed and more accurate results on the variations of the pressure, temperature, partial densities, and enthalpy as a function of time.

Schlieren cinematographic and holographic diagnostic of a laser-produced plasma in xenon

We have studied extensively a gas breakdown in xenon produced by a giant pulse ruby laser with a power <100 MW. Detailed information concerning the structure of the laser plasma formation and of the following expansion has been obtained by different optical methods, including schlieren techniques (single frames and ultra high speed cinematography) and holography. The high quality of the holographic pictures was achieved by the use of a monomode laser. With this we were able to visualize the plasma history and to determine the velocities of the boundary layer and of the blast wave with utmost accuracy. Investigations of the electron density and electron temperature showed that a relaxation time of about 10 nsec is necessary to establish local thermodynamic equilibrium states. After this relaxation time it is then possible to carry out thermodynamic calculations, applying the shock-wave theory, to relate the optically measured expansion velocity with the plasma parameters involved. The mean specific internal energy ∊¯, for instance, attained values in excess of 1012 erg/g which decayed rapidly during the first 100 nsec to about 5 · 1010 erg/g. By comparing the results to theoretical calculations of ∊¯ we obtained a first estimate of the temperature, taking into consideration the partial densities nj as well as the partition functions Zj(i) of the xenon atoms, the single-charged ions, and the double-charged ones. Furthermore, a two-step iteration computer program was used to give more detailed and more accurate results on the variations of the pressure, temperature, partial densities, and enthalpy as a function of time.