Constant Power Model in Arm Rotation—A New Approach to Hill’s Equation

The purpose of this study was to further develop the constant power model of a previous study and to provide the final solution of Hill’s force-velocity equation. Forearm and whole arm rotations of three different subjects were performed downwards (elbow and shoulder extension) and upwards (elbow and shoulder flexion) with maximum velocity. These arm rotations were recorded with a special camera system and the theoretically derived model of constant maximum power was fitted to the experimentally measured data. The moment of inertia of the arm sectors was calculated using immersion technique for determining accurate values of friction coefficients of elbow and whole arm rotations. The experiments of the present study verified the conclusions of a previous study in which theoretically derived equation with constant maximum power was in agreement with experimentally measured results. The results of the present study were compared with the mechanics of Hill’s model and a further development of Hill’s force-velocity relationship was derived: Hill’s model was transformed into a constant maximum power model consisting of three different components of power. It was concluded that there are three different states of motion: 1) the state of low speed, maximal acceleration without external load which applies to the hypothesis of constant moment;2) the state of high speed, maximal power without external load which applies to the hypothesis of constant power and 3) the state of maximal power with external load which applies to Hill’s equation. This is a new approach to Hill’s equation.

Vibrational Stabilization by Reshaping Arnold Tongues: A Numerical Approach

This paper presents two contributions to the stability analysis of periodic systems modeled by a Hill equation: The first is a new method for the computation of the Arnold Tongues associated to a given Hill equation which is based on the discretization of the latter. Using the proposed method, a vibrational stabilization is performed by a change in the periodic function which guarantees stability, given that the original equation has unbounded solutions. The results are illustrated by some examples.

Study of Linearization of Hill Dose-Effect Curve with Metabolic Velocity Instead of Drug Concentration

Objective To explore the velocity-effect relationship in order to the establish linearization of effect on an equation with regard to the consistency of the Hill dose-effect expression with the metabolic kinetics of receptors.Methods The linear velocity-effect expression was obtained by solving multivariant differential equation groups,which were established to compare the coincidences and basic relations between the Hill dose-effect and metabolic kinetic Michaelis-Menten equation for receptors.The validation test was conducted with acetylcholine,adrenaline,and their mixture as model drugs.Results The linear velocity-effect modelling was represented in vivo or in vitro,for single and multidrug systems.Pharmacodynamic parameters,especially suitable for multicomponent CMM formulas,could be determined and calculated for single or multicomponent formulas at high saturating or low linear concentration for receptors.The validation test showed that the pharmacodynamic parameters of acetylcholine were:k,2.675×10^-3s^-1;ka,5.786×10^-9s^-1;km,2.500×10^-7s^-1;α,4.619×10^9张s·mg^-1;E0,13张(P<0.01)and those of adrenaline were:k,1.415×10^-3s^-1;ka,5.846×10^-9s^-1;km,2.300×10^-7s^-1;α,-1.627×10^9张s·m g^-1;E0,9.2张(P<0.01).For the mixture of the two components,the values were:α,1.375×1010张s·m g^-1;-6.150×10^9张s m g^-1for acetylcholine and adrenaline,respectively,and E0was7.08张in both,with the other parameters unchanged(P<0.01).Conclusion The velocity-effect equation can linearize the Hill dose-effect relationship,which can be applied to study the pharmacodynamics and availability of CMM formulations in vivo and in vitro.

OSCILLATION OF A CLASS OF HILLE EQUATION

This paper studies a class of Hille equation. A formula for solutions of a class of Hille equation is given. Under some suitable conditions the oscillation and nonoscillation of a class of Hille equation are established. Our results generalize the known Hille＇s ones.

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill’s equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.