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Given a function f : X → Y of metric spaces, its asymptotic dimension asdim(f ) is the supremum of asdim(A) such that A ⊂ X and asdim(f (A)) = 0. Our main result is generalizes a result of Bell and Dranishnikov [3] in which f isLipschitz and X is geodesic. We provide analogs of 0.1 for Assouad-Nagata dimension dim AN and asymptotic Assouad-Nagata dimension asdim AN . In case of linearly controlled asymptotic dimension l-asdim we provide counterexamples to three questions of Dranishnikov [14].As an application of analogs of 0.1 we prove Theorem 0.2. If 1 → K → G → H → 1 is an exact sequence of groups and G is finitely generated, then asdim AN (G, d G ) ≤ asdim AN (K, d G |K) + asdim AN (H, d H ) for any word metrics metrics d G on G and d H on H. 0.2 extends a result of Bell and Dranishnikov [3] for asymptotic dimension.

Permeability-surface area products of the capillary wall, PSc, and the myocyte sarcolemma, PSpc, for D-glucose and 2-deoxy-D-glucose were estimated via the multiple indicator-dilution technique in isolated blood-perfused dog and Tyrode-perfused rabbit hearts. Aortic bolus injections contained 131I-albumin (intravascular reference), two of three glucoses: L-glucose (an extracellular reference solute), D-glucose, and 2-deoxy-D-glucose. Outflow dilution curves were sampled for 1-2.5 min without recirculation. The long duration sampling allowed accurate evaluation of PSpc by fitting the dilution curves with a multiregional axially distributed capillary-interstitial fluid-cell model accounting for the heterogeneity of regional flows (measured using microspheres and total heart sectioning). With average blood flow of 1.3 ml . g-1 . min-1, in the dog hearts the PSc for D-glucose was 0.72 +/- 0.17 ml . g-1 . min-1 (mean +/- SD; n = 11), and PSpc was 0.57 +/- 0.15 ml . g-1 . min-1. In the rabbit hearts with perfusate flow of 2.0 ml . g-1 . min-1 (n = 6), PSc was 1.2 +/- 0.1 and PSpc was 0.4 +/- 0.1 ml . g-1 . min-1. PSc for 2-deoxy-D-glucose was about 4% higher than for D-glucose and L-glucose in both preparations. Relative to L-glucose, there was no measurable transendothelial transport of either dextroglucose, indicating that transcapillary transport was by passive diffusion, presumably via the clefts between cells. The technique allows repeated measurements of D-glucose uptake at intervals of a few minutes; it may therefore be used to assess changes in transport rates occurring over intervals of several minutes.

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