On Some Properties of Functions on Convex Galaxies

In this paper, we define and study extensively a new type of external sets in R , we call it "convex galaxies". We show that these convex external sets may be classified in some definite types. More precisely, we obtain the following : (1) Let R G  be a convex galaxy which is symmetric with respect to zero, then (i) G is an  galaxy (0) if and only if there exists an internal strictly increasing sequence of strictly positive real numbers   N n n a  with   n n N n a a G ,     such that   0 a and c a a n n  1 , for all N n , where, c is some limited real number such that 1  c . Tahir H. Ismail Barah M. Sulaiman & Hind Y. Saleh 166 (ii) G is a non-linear galaxy if and only if there exists an internal strictly increasing sequence of strictly positive real numbers   N n n a  with   n n N n a a G ,     such that n n a a 1  is unlimited for all N n . (2) Let R G  be a convex galaxy which is symmetric with respect to zero, then (i) G is an  galaxy (0) iff there exists a real internal strictly increasing  C function f , such that G G f  ) ( , and c t f t f   ) ( ) ( for all limited 1  t , where c is a positive real number. (ii) G is a non-linear galaxy if and only if there exists a real internal strictly increasing  C function f , such that G G f  ) ( and ) ( ) ( t f t f  is positive unlimited, for all appreciable 1  t .

(2) Let R G  be a convex galaxy which is symmetric with respect to zero, then

Introduction
An application of this classification may be found in non-standard analysis approach. The study of slow-fast vector fields as shown by Diener F. [2]. For example, the notion width of jump may be defined in terms of a convex galaxy. Further, this classification can be used in approximations. Thus, the set of points on the real line, where two real functions f and g are infinitely close on R , that is the set will often be a convex monad. [1] For practice reasons we start with the study of convex galaxies which are symmetric with respect to zero.
Every concept concerning sets or elements defined in the classical mathematics is called standard.
Any set or formula which does not involve new predicates "standard, infinitesimals, limited, unlimited … etc" is called internal, otherwise it is called external.
A real number x is called unlimited if and only if xr  for all positive standard real numbers r ; otherwise it is called limited. The set of all unlimited real numbers is denoted by R , and the set of all limited real numbers is denoted by R .

A real number
x is called infinitesimal if xr  for all positive standard real numbers r . A real number x is called appreciable, if x is limited but not infinitesimal. Two real numbers x and y are said to be infinitely close if and only if xy − is infinitesimal and denoted by y x  . The set of all limited real numbers is called principal galaxy, (denoted by ).

For any real number a , the set of all real numbers
x such that xa − limited is called the galaxy of a (denoted by gal(a)).
Let ( ) 0    and xR  ,we define the α-galaxy (x) as follows: α-galaxy(x)={ : yx yR  −  is limited} and denoted by α-G(x). A subset G of R is a convex galaxy which is asymmetric with respect to zero iff there exists an internal strictly increasing sequence of strictly positive real numbers

Theorem 1.1:( Extension Principle) [3]
Let X and Y be two standard sets, s X and s Y be the subsets constitute of the standard elements X and Y, respectively. If we can associate with every , then there exists a unique standard * yY  such that st x  s X ,

Theorem 1.2: (Principal of External Induction) [3]
If E is an internal or external property such that E(0) is true and Since, 0 1 a = , it follows that G= . If, on the contrary, we had 0 a  = , then G will be α- G  . A convex galaxy which is a group, but not an α-galaxy will be called non-linear. Informally, the α-galaxy is the set of all real numbers of order α, while a non-linear galaxy cannot be the set of real numbers of the order of one of its element.
We generalize the connection between the convex galaxy G and the ratio of Hence, α-galaxy(x) G