Free entropy and property T factors

We show that a large class of finite factors has free entropy dimension less than or equal to one. This class includes all prime factors and many property T factors.

Multiscale assessment of patterns of avian species richness

The search for a common cause of species richness gradients has spawned more than 100 explanatory hypotheses in just the past two decades. Despite recent conceptual advances, further refinement of the most plausible models has been stifled by the difficulty of compiling high-resolution databases at continental scales. We used a database of the geographic ranges of 2,869 species of birds breeding in South America (nearly a third of the world's living avian species) to explore the influence of climate, quadrat area, ecosystem diversity, and topography on species richness gradients at 10 spatial scales (quadrat area, ≈12,300 to ≈1,225,000 km2). Topography, precipitation, topography × latitude, ecosystem diversity, and cloud cover emerged as the most important predictors of regional variability of species richness in regression models incorporating 16 independent variables, although ranking of variables depended on spatial scale. Direct measures of ambient energy such as mean and maximum temperature were of ancillary importance. Species richness values for 1° × 1° latitude-longitude quadrats in the Andes (peaking at 845 species) were ≈30–250% greater than those recorded at equivalent latitudes in the central Amazon basin. These findings reflect the extraordinary abundance of species associated with humid montane regions at equatorial latitudes and the importance of orography in avian speciation. In a broader context, our data reinforce the hypothesis that terrestrial species richness from the equator to the poles is ultimately governed by a synergism between climate and coarse-scale topographic heterogeneity.

Persistent confusion of total entropy and chemical system entropy in chemical thermodynamics.

The change in free energy with temperature at constant pressure of a chemical reaction is determined by the sum (dS) of changes in entropy of the system of reagents, dS(i), and the additional entropy change of the surroundings, dS(H), that results from the enthalpy change, W. A faulty identification of the total entropy change on reaction with dS(i) has been responsible for the attribution of general validity to the expressions (d deltaG/dT)p = -deltaS(i) and d(deltaG/T)/d(1/T)= deltaH, which are found in most textbooks and in innumerable papers.