Thermal conductivity of gaseous and liquid hydrogen

Normal and para-hydrogen conductivity measurements at temperatures from 200 to 17 deg K, at densities up to 2.6 times critical density, and at pressures to 15 MN/sq m are made. Using new calorimeter, data are analyzed as functions of density at fixed temperatures and of temperature at fixed densitie

Thermodynamic and transport properties of fluids and selected solids for cryogenic applications Summary report, 1 Dec. 1965 - 1 Nov. 1970

Summary data on thermodynamic and transport properties of fluids and solids for cryogenic application

Introduction

Author Institution: Director, Natural Rescources Institute, The Ohio State University, Columbus 10 ; Chariman, Department of Forestry, Ohio Agricultural Experiment Station, Wooster, Ohi

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure