Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Nonlinear indicial response of complex nonstationary oscillations as pulmonary hypertension responding to step hypoxia

This paper is devoted to the quantization of the degree of nonlinearity of the relationship between two biological variables when one of the variables is a complex nonstationary oscillatory signal. An example of the situation is the indicial responses of pulmonary blood pressure (P) to step changes of oxygen tension (ΔpO2) in the breathing gas. For a step change of ΔpO2 beginning at time t1, the pulmonary blood pressure is a nonlinear function of time and ΔpO2, which can be written as P(t-t1 | ΔpO2). An effective method does not exist to examine the nonlinear function P(t-t1 | ΔpO2). A systematic approach is proposed here. The definitions of mean trends and oscillations about the means are the keys. With these keys a practical method of calculation is devised. We fit the mean trends of blood pressure with analytic functions of time, whose nonlinearity with respect to the oxygen level is clarified here. The associated oscillations about the mean can be transformed into Hilbert spectrum. An integration of the square of the Hilbert spectrum over frequency yields a measure of oscillatory energy, which is also a function of time, whose mean trends can be expressed by analytic functions. The degree of nonlinearity of the oscillatory energy with respect to the oxygen level also is clarified here. Theoretical extension of the experimental nonlinear indicial functions to arbitrary history of hypoxia is proposed. Application of the results to tissue remodeling and tissue engineering of blood vessels is discussed.