Contents 1. Introduction 597 2. Braid groups in positive genus 601 3. Relative completion of mapping class groups 603 4. Mixed Hodge structures on Torelli groups 608 5. Review of continuous cohomology 613 6. Remarks on the representations of sp g 616 7. Continuous cohomology of Torelli groups 618 8. The lower central series quotients of a surface group 621 9. The action of t 1 g on p g 623 10. ...

Let (U, R) be an approximation space with U being non-empty set and R equivalence relation on U, let $${\overline{G}}$$ $${\underline{G}}$$ the upper lower of subset G U. A topological rough group is a $$G=({\underline{G}}, {\overline{G}})$$ endowed topology, which induced from , such that product mapping $$f: G\times G\rightarrow {\overline{G}}$$ inverse are continuous. In class groups, relati...

The Fourier Transform is an important mathematical tool in a wide variety of fields of science and engineering [1, p. XI]. In this paper, by the use of some integral identities and inequalities developed in [2](see also [3]), we point out some approximations of the Fourier transform in terms of the complex exponential mean, E (z, w) (see Section 2) and study the error of approximation for diffe...

This study introduce a new type of closed sets in topology called Generalized h-closed (briefly, gh-closed) define as follow: E ⊆ χ be gh-closed set if CLh (E)⊆ U whenever and is open (χ,τ). The relation between other classes ( h-closed, g-closed, g -closed, g-closed αg-closed) are studied. Also, the notion gh-continuous mapping on topological space some properties proved. Finally, separation a...

and a0 is a constant, b is a continuous periodic function of variables x and y, a1 and g are continuous periodic functions. This integral equation appears in exterior boundary value problems for the two-dimensional Helmholtz equation (see [11,15– 17,27]). Constructions of conformal mapping also lead to integral equations of this kind (see [22]). In recent years, the Helmholtz equation and its n...

it is well known that a microperiodic function mapping a topological group into reals, which is continuous at some point is constant. we introduce the notion of a microperiodic multifunction, defined on a topological group with values in a metric space, and study regularity conditions implying an analogous result. we deal with vietoris and hausdorff continuity concepts.stability of microperiodi...