The index growth and multiplicity of closed geodesics

In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in \cite{LoD1} on rational and in \cite{DuL1}, \cite{Rad4} and \cite{Rad5} on completely non-degenerate closed geodesics on spheres and \CP^2 to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.Comment: 59 pages, 1 figure, to appear in Journal of Functional Analysis (JFA) and this is the final versio

Synthesis, assembly, and architecture of titanium dioxide based semiconductor nanostructures

Ph.DDOCTOR OF PHILOSOPH

Myocardial Electrophysiological Alterations In Pacing-induced Heart Failure And Defibrillation Shock Delivery

Heart failure patients frequently have ventricular arrhythmias and high incidence of sudden death. Current antiarrhythmic drug therapies have failed to improve the survival of these patients as the arrhythmogenesis is poorly understood. The only effective therapy for lethal arrhythmia, ventricular fibrillation, is electrical shock (defibrillation). Studies on arrhythmogenesis and potential antiarrhythmic therapies in heart failure are needed.;This thesis investigated: (1) cardiac electrophysiological changes and arrhythmogenesis in a heart failure model; (2) cellular electrophysiological effects of defibrillation shocks.;Heart failure was produced by rapid ventricular pacing (240 beats/min, 4-5 weeks) in 35 dogs. Twenty-one operated, non-paced dogs served as control. Increased impulse conduction time, ventricular refractory period and action potential duration, and decreased action potential duration and resting membrane potential of Purkinje fibers were observed in association with heart failure. Despite a high incidence of ventricular fibrillation during heart failure, programmed electrical stimulation failed to induce significant arrhythmia, suggesting a non-reentrant mechanism for the arrhythmia initiation. The arrhythmogenic effect of norepinephrine and myocardial responsiveness to isoproterenol were both decreased, suggesting a decreased importance of catecholamines in the arrhythmogenesis of heart failure. The failing heart showed an increased susceptibility to cesium-induced ventricular tachycardia and triggered activity, implying a possible important role for triggered activity in the arrhythmogenesis of heart failure.;Defibrillation shocks to guinea pig papillary muscle caused refractory states which were related to shock intensity and waveforms. This refractoriness lasted longer than one activation and may be important for successful defibrillation. Shocks also increased stimulation threshold, which may contribute to post-defibrillation pacing failure in patients with an implanted pacemaker. In canine Purkinje fibers, shocks induced rapid firing, which may be responsible for unsuccessful defibrillation and post-defibrillation arrhythmias.;Conclusions: (1) Heart failure is associated with arrhythmogenic electrophysiological abnormalities and increased susceptibility to ventricular fibrillation. The arrhythmogenesis is unlikely to be mainly due to reentry or catecholamines whereas triggered activity may play an important role. (2) Defibrillation shock-induced refractoriness lasts longer than one activation, which may be important for successful defibrillation. The shock-induced increase of pacing threshold and rapid firing may constitute potential complications of defibrillation, including reinitiation of ventricular fibrillation