Relative Predicativity and dependent recursion in second-order set theory and higher-orders theories

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

Potassium and the excitability properties of normal human motor axons in vivo.

Hyperkalemia is an important cause of membrane depolarization in renal failure. A recent theoretical model of axonal excitability explains the effects of potassium on threshold electrotonus, but predicts changes in superexcitability in the opposite direction to those observed. To resolve this contradiction we assessed the relationship between serum potassium and motor axon excitability properties in 38 volunteers with normal potassium levels. Most threshold electrotonus measures were strongly correlated with potassium, and superexcitability decreased at higher potassium levels (P = 0.016), contrary to the existing model. Improved modelling of potassium effects was achieved by making the potassium currents obey the constant-field theory, and by making the potassium permeabilities proportional to external potassium, as has been observed in vitro. This new model also accounted well for the changes in superexcitability and other excitability measures previously reported in renal failure. These results demonstrate the importance of taking potassium levels into account when assessing axonal membrane dysfunction by excitability testing, and provide evidence that potassium currents are activated by external potassium in vivo.

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.