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Scientia Magna : An International Journal : Volume 2, No. 4, 2006

By Xi'an, Shaanxi

Book Id:WPLBN0002828570 Format Type:PDF eBook: File Size:3.55 MB Reproduction Date:8/7/2013

Xi'anitor, B. S. (Ed.). (2013). Scientia Magna : An International Journal : Volume 2, No. 4, 2006. Retrieved from http://gutenberg.us/

Description
Scientia Magna is published annually in 200-300 pages per volume and 1,000 copies on topics such as mathematics, physics, philosophy, psychology, sociology, and linguistics.

Excerpt
Smarandache inversion sequence
Abstract We study the Smarandache inversion sequence which is a new concept, related sequences, conjectures, properties, and problems. This study was conducted by using (Maple 8){a computer Algebra System.
Keywords Smarandache inversion, Smarandache reverse sequence.
Introduction
In [1], C.Ashbacher, studied the Smarandache reverse sequence:
1; 21; 321; 4321; 54321; 654321; 7654321; 87654321; 987654321; 10987654321; 1110987654321; (1) and he checked the _rst 35 elements and no prime were found. I will study sequence (1), from different point of view than C. Ashbacher. The importance of this sequence is to consider the place value of digits for example the number 1110987654321, to be considered with its digits like this : 11; 10; 9; 8; 7; 6; 5; 4; 3; 2; 1, and so on. (This consideration is the soul of this study because our aim is to study all relations like this (without loss of generality).
Definition. The value of the Smarandache Inversions (SI) of a positive integers, is the number of the relations i > j ( i and j are the digits of the positive integer that we concern with it), where i always in the left of j as the case of all numbers in (1). I will study the following cases of above equation.
Examples. The number 1234 has no inversions ((SI) = 0, or zero inversion), also the number 1, while the number 4321 has 6 inversions, because 4 > 3 > 2 > 1, 3 > 2 > 1, and 2 > 1. The number 1110987654321 has 55 inversions, and 1342 has two inversions. So our interest will be of the numbers in Smarandache reverse sequence i.e. (1), because it has mathematical patterns and interesting properties.

Table of Contents
M. Karama : Smarandache inversion sequence 1
W. He, D. Cui, Z. Zhao and X. Yan : Global attractivity of a recursive sequence 15
J. Earls : Smarandache reversed slightly excessive numbers 22
X. Ren and H. Zhou : A new structure of super R¤-unipotent semigroups 24
J. Li : An equation involving the Smarandache-type function 31
P. Zhang : An equation involving the function ±k(n) 35
Y. B. Jun : Smarandache fantastic ideals of Smarandache BCI-algebras 40
T. Jayeo. la : On the Universality of some Smarandache loops of Bol-Moufang type 45
W. Kandasamy, M. Khoshnevisan and K. Ilanthenral : Smarandache representation and its applications 59
Y. Wang, X. Ren and S. Ma : The translational hull of superabundant semigroups with semilattice of idempotents 75
M. Bencze : Neutrosophic applications in ¯nance, economics and politics|a continuing bequest of knowledge by an exemplarily innovative mind 81
J. Fu : An equation involving the Smarandache function 83
W. Zhu : The relationship between Sp(n) and Sp(kn) 87
X. Du : On the integer part of the M-th root and the largest M-th power not exceeding N 91
J. Earls : On the ten's complement factorial Smarandache function 95
B. Shi : The hybrid mean value of the Smarandache function and the Mangoldt function 98
H. Ibstedt : Palindrome studies (Part I) The palindrome concept and its applications to prime numbers 101