INTERPOLATING BLASCHKE PRODUCTS AND ANGULAR DERIVATIVES

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra H-infinity[(b) over bar : b has finite angular derivative everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Terrestrial feeding in aquatic turtles: environment-dependent feeding behavior modulation and the evolution of terrestrial feeding in Emydidae

Evolutionary transitions between aquatic and terrestrial environments are common in vertebrate evolution. These transitions require major changes in most physiological functions, including feeding. Emydid turtles are ancestrally aquatic, with most species naturally feeding only in water, but some terrestrial species can modulate their feeding behavior appropriately for both media. In addition, many aquatic species can be induced to feed terrestrially. A comparison of feeding in both aquatic and terrestrial environments presents an excellent opportunity to investigate the evolution of terrestrial feeding from aquatic feeding, as well as a system within which to develop methods for studying major evolutionary transitions between environments. Individuals from eight species of emydid turtles (six aquatic, two terrestrial) were filmed while feeding underwater and on land. Bite kinematics were analyzed to determine whether aquatic turtles modulated their feeding behavior in a consistent and appropriate manner between environments. Aquatic turtles showed consistent changes between environments, taking longer bites and using more extensive motions of the jaw and hyoid when feeding on land. However, these motions differ from those shown by species that naturally feed in both environments and mostly do not seem to be appropriate for terrestrial feeding. For example, more extensive motions of the hyoid are only effective during underwater suction feeding. Emydids evolving to feed on land probably would have needed to evolve or learn to overcome many, but not all, aspects of the intrinsic emydid response to terrestrial feeding. Studies that investigate major evolutionary transitions must determine what responses to the new environment are shown by naïve individuals in order to fully understand the evolutionary patterns and processes associated with these transitions.

Interpolating Blaschke Products and Angular Derivatives

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra generated by the algebra of bounded analytic functions and the conjugates of Blaschke products with angular derivative finite everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.