In this paper, we apply a successful combination of three key tools which allows to get a measure of the cost of the approximate controllability for semilinear heat equation. The first tool consists to get enough information about the approximate control for the linear heat equation with a potential depending on space-time variable. Then a fixed point method is applied. The fixed point techniqu...

This is the second of three lectures on introduction to vertex operator algebras. In this lecture, we shall continue Professor Dong’s lecture to present more fundamental properties of vertex operator algebras. From the mathematical point of view, a vertex operator algebra formally resembles a Lie algebra because the Jacobi identity is used as one of the main axioms. For the Lie algebra aspect o...

BACKGROUND The frizzled (fz) gene of Drosophila encodes the founding member of the large family of receptors for the Wnt family of signaling molecules. It was originally studied in the adult epidermis, where it plays a key role in the generation of tissue polarity. Mutations in components of the fz signal transduction pathway disrupt tissue polarity; on the wing, hairs normally point distally b...

In this paper, we study a fixed-point problem with set-valued mapping by using an algorithm based on unions of nonexpansive mappings. We show that approximate solution is reached after finite number iterations in the presence computational errors. This result extension results known literature.

objective: the aim of the present study was to retrospectively evaluate small-diameter (3.3 mm) straumann® dental implants placed in the maxilla or the mandible over a period of 5 years in function. materials and methods: twenty- eight partially edentulous patients received a total of 48 implants over a 5-year period. after the standard healing period (3 to 6 months), the implants were restored...