The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].

Abstract. We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms and establish some of their fundamental properties. We give much more detailed results about their structure in the case that there are only a finite number of generators and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if whenever X is a Banach lattice, J a closed ideal in X, Q : X → X/J the quotient map, T : P → X/J a linear lattice homomorphism and > 0 there is a linear lattice homomorphismT : P → X such that (i) T = Q •T and (ii)T ≤ (1 + ) T . We establish the connection between projective Banach lattices and free Banach lattices and describe several families of Banach lattices that are projective as well as proving that some are not.

The effect of a constant magnetic field, applied at an angle P to the easy axis of magnetization, on the Neel relaxation time r of a single domain ferromagnetic particle (with uniaxial anisotropy) is studied by calculating the lowest nonvanishing eigenvalue k, (the escape rate) of the appropriate Fokker-Planck equation using matrix methods. The effect is investigated by plotting kl versus the anisotropy parameter n for various values of P, and the ratio h =g/2a t where g is the external field parameter and k& versus P for various h values (for rotation of the magnetization vector M both in a plane and in three dimensions). If M rotates in a plane the curve of k& versus 1/I is symmetric about t/t=vrl4 in the range 0(t/1(m/2 and significant decrease in r with increasing 1/I is predicted for large ( and n The m. aximum decrease in r occurs at t/mI/4 whereupon r increases again to the /=0 value at P=rr/2. For rotation of M in three dimensions, the curve of X, versus P (0(~7r) is symmetric about~7r/2. Thus the maximum decrease in r again occurs at P= 7r/4 with maximum increase to a value exceeding that at /=0 (i.e., with the field applied along the polar axis with that axis taken as the easy axis), at t/r=rr/2 (field applied along the equator), the /=0 value being again attained at $=7r The.results are shown to be consistent with the behavior predicted by the Kramers theory of the rate of escape of particles over potential barriers. This theory when applied to the potential barriers for the equatorial orientation of the field for rotation in three dimensions yields a simple approximate formula for the escape rate which is in reasonable agreement with the exact k& calculated from the Fokker-Planck equation. Pfeiffer's approximate formula for the barrier height as a function of tr [H. Pfeiffer, Phys. Status Solidi 122, 377 (1990)]is shown to be in reasonable agreement with our results.

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