We study the relations of being substructure and elementary substructure between Kripke models of intuitionistic predicate logic with the same arbitrary frame. We prove analogues of Tarski’s test and Löwenheim-Skolem’s theorems as determined by our definitions. The relations between corresponding worlds of two Kripke models K K′ are studied. —————————————————————————2000 Mathematics Subject Cla...

The observables in a strong gravitational lens are usually just the image positions and sometimes the flux ratios. We develop a new and simple algorithm which allows a set of models to be fitted exactly to the observations. Taking our cue from the strong body of evidence that early-type galaxies are close to isothermal, we assume that the lens is scale-free with a flat rotation curve. However, ...

Multiply-imaged quasar lenses can be used to constrain the substructure mass fraction in galaxysized dark matter halos via anomalous flux ratios in lensed images. The flux ratios, however, can be affected by both the substructure in the lens halo and by isolated small-mass halos along the entire line-of-sight to the lensed source. While lensing by dark matter clumps near the lens galaxy is more...