Non-Solvable Equation Systems with Graphs Embedded in Rn

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International J.Math. Combin. Vol.2(2013), 08-23

Non-Solvable Equation Systems with Graphs Embedded in Rn Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China) E-mail: maolinfan@163.com

Abstract: Different from the homogenous systems, a Smarandache system is a contradictory system in which an axiom behaves in at least two different ways within the same system, i.e., validated and invalided, or only invalided but in multiple distinct ways. Such systems widely exist in the world. In this report, we discuss such a kind of Smarandache system, i.e., non-solvable equation systems, such as those of non-solvable algebraic equations, non-solvable ordinary differential equations and non-solvable partial differential equations by topological graphs, classify these systems and characterize their global behaviors, particularly, the sum-stability and prod-stability of such equations. Applications of such systems to other sciences, such as those of controlling of infectious diseases, interaction fields and flows in network are also included in this report.

Key Words: Non-solvable equation, Smarandache system, topological graphs, vertex-edge labeled graph, G-solution, sum-stability, prod-stability.

AMS(2010): 05C15, 34A30, 34A34, 37C75, 70F10, 92B05 §1. Introduction Consider two systems of linear equations following:   x+y      x+y (LES4N )   x−y     x−y

=

1

=

−1

=

−1

=

1

  x=y      x+y =2 (LES4S )   x=1     y=1

Clearly, (LES4N ) is non-solvable because x + y = −1 is contradictious to x + y = 1, and so that for equations x − y = −1 and x − y = 1. Thus there are no solutions x0 , y0 hold with all equations in this system. But (LES4S ) is solvable clearly with a solution x = 1 and y = 1. It should be noted that each equation in systems (LES4N ) and (LES4S ) is a straight line in Euclidean space R2 , such as those shown in Fig.1. 1 Reported at the First International Conference on Smarandache Multispaces and Multistructures, June 28-30,2013, Beijing, P.R.China. 2 Received April 6, 2013, Accepted June 6,2013.


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