b-PARTS OF REAL NUMBERS AND THEIR GENERALIZATION
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Date
2010-03
Authors
Hooshmand, Mohamad Hadi
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Abstract
Considering the basic and main properties of the b-parts of real numbers, we have
studied them in three aspects: elementary, analytic and 'algebraic methods.
As the first aspect we have several subjects and their applications such as:
(a) Number theoretic explanations of b-parts and their elementary properties.
(b) Application of b-parts for unique finite and infinite representation of real numbers,
applying them and the generalized division algorithm, we not only introduce some
direct formulas for digits of the unique infinite expansion of real numbers to the base
an integer but also prove a (new) unique finite representation of real numbers to the
base an arbitrary real number (not equal 0, ±1).
(c) Application of b-parts for determining the general form of b-periodic real subsets
and functions.
As for analytic aspect, we consider some asymptotic and direct formulas for the
partial summation of b-parts and remainders of the generalized divisions of a given
positive real number. Also we}ntend to explain the conception of uniform distribution
of real sequences modulo b (for an arbitrary real number b =f 0) by using the b-decimal
part function, as an application of the b-parts.
As the third aspect we introduce b-addition of real numbers that is b-decimal part
of their ordinary addition and "the least real b-residues group" (b-bounded group)
and study their properties as well as relations to real groups and their quotient groups.
Generalization of b-parts for arbitrary groups is another topic that we study. In
this way we show how an element of a group can be uniquely represented by cyclic
and f-part, like b-integer and b-decimal part of real numbers. This consideration will
give us the generalized division algorithm for Abelian groups with no element of the
finite order. In this part of our research we also focus on decomposer and associative
functions on groups (even on semi-groups and binary systems) and solve the related
functional equations.
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Keywords
NUMBERS , GENERALIZATION