We introduce majorant computability of functions on reals. A structural theorem is proved, which connects the graph of a majorant{computable function with validity of a nite formula on the set of the hereditarily nite set HF( R) (where R is an elementary proper extension of standard real numbers). The class of majorant{computable functions in our approach include an interesting class of real to...

Characteristic properties of majorant-computable real-valued functions are studied. A formal theory of computability over the reals which satisses the requirements of numerical analysis used in Computer Science is constructed on the base of the deenition of majorant-computa-bility proposed in 13]. A model-theoretical characterization of majorant-computability real-valued functions and their dom...

Contributions to the Theory of Optimal Stopping for One–Dimensional Diffusions Savas Dayanik Advisor: Ioannis Karatzas We give a new characterization of excessive functions with respect to arbitrary one–dimensional regular diffusion processes, using the notion of concavity. We show that excessive functions are essentially concave functions, in some generalized sense, and vice–versa. This, in tu...

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new...