No abstract

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical GJMS operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order −2, originally defined by A. Connes, acting on middle dimension forms.

Abstract. On a 6-dimensional, conformal, oriented, compact manifold M without boundary, we compute a whole family of differential forms Ω 6

Resumen. La principal intención de este trabajo es motivar a los docentes e investigadores en educación matemática a integrar en los procesos de enseñanza y aprendizaje de las matemáticas relacionados con el concepto de función, el desarrollo histórico de dicho objeto de estudio. Como segundo objetivo se desea sugerir diferentes actividades que se pueden utilizar para estudiar el concepto de función en los varios niveles de la educación formal. Este artículo se divide en tres secciones. La primera sección es una revisión del desarrollo del concepto de función a través de la historia. La segunda sección es un breve estudio de los tipos de definición existentes y las diferentes formas de representar funciones. La tercera sección es un recuento de actividades o situaciones de interés, con la intención de indicar facetas interesantes a la hora de estudiar el concepto de función.Palabras clave: historia de las matemáticas, función, actividades de enseñanza aprendizaje.Abstract. The main intention of this work is to motivate teachers and researchers in math education to integrate in the processes of teaching and learning mathematics related to the concept of function, the historical development of such an object of study. As a second objective, it is intended to suggest different activities that can be used to study the concept of function in the various levels of formal education. The first of three sections in this paper is a historical review of the development of the concept of function . The second section is a brief study of the types of existing definitions and the different ways to represent a function. The third section lists activities of situations of interest, aiming to identify interesting stages during the study of the concept of function.

We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the setting of compact manifolds with boundary.Furthermore, in the 4-dimensional case, Connes has also shown that the Paneitz operator [14] (critical GJMS for n = 4 [10]), can be derived from B 4 by the relationAiming to extend the work of Connes to even dimensional manifolds, in [18] we have proved the following two results:Theorem 1 of [18]. Let M be an n-dimensional compact conformal manifold without boundary. Let S be a pseudodifferential operator of order 0 acting on sections of a vector bundle over M such that S 2 f 1 = f 1 S 2 and the pseudodifferential operator P = [S, f 1 ][S, f 2