The current definition of the derivative makes the set of differentiable functions much smaller than the set of continuous functions, such that most of the real single variable functions are not differentiable and the surveying the rate of their growth is not possible with the available definition. In the present paper, using Caratheodory definition, we extend the set of differentiable functions by introducing a definition for generalized differentiation of a single variable function and its generalized derivative, in such a way that the validity of the basic theorems of this theory such as Rolle's theorem, Cauchy's mean value theorem, mean value theorem and Taylor's theorem would be hold. Finally we give some examples.

The concepts of L-convex systems and Scott-hull spaces are pro-posed on frame-valued setting. Also, we establish the categorical isomorphism between L-convex systems and Scott-hull spaces. Moreover, it is proved that the category of L-convex structures is bireflective in the category of L-convex systems. Furthermore, the quotient systems of L-convex systems are studied.

Let X be a Banach space of functions analytic on a plane domain  W such that for every l in W the functional of evaluation at l is bounded. Assume further that X contains the constants and admits multiplication by the independent variable z, Mz, as a bounded operator. We give sufficient conditions for Mz to be reflexive.

In this paper, first, we present a kinetic data structure for the problem of kinetic convex hull maintenance. After that, discuss some complicated cases followed by the main algorithm and two additional algorithms. Finally, focus on the evaluation of the four properties of the kinetic algorithm in the worst case. We show that quality factors of our algorithm are at most linear. Comparing with similar work, our suggestion is easier to implement and keep track of.

Let H be a Hilbert space of functions analytic on a plane domain W such that for every l in W the functional of evaluation at l is bounded. Assume further that H contains the constants and admits multiplication by the independent variable z, Mzn as a bounded operator. We give sufficient conditions for Mz to be reflexive for all positive integers.

Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational ge-ometry and other areas like image processing, model identi cation, geographical data systems, and triangu-lar computation of a set of points and so on. Comput-ing the convex hull of a set of point is one of the most fundamental and important problems of computational geometry. In this paper a new algorithm is presented for computing the convex hull of a set of random points in the plane by using a sweep-line strategy. The sweep-line is a horizontal line that is moved from top to bottom on a map of points. Our algorithm is optimal and has time complexity O(nlogn) where n is the size of input.