# International Journal of Mathematical Combinatorics, international book series, Vol. 2, 2012

For such systems, the linear equation systems without solutions, i. We characterize such non- solvable linear equation systems with their homeomorphisms, particularly, the non-solvable linear equation systems with 2 or 3 variables by combinatorics. It is very interesting that every planar graph with each edge a straight segment is homologous to such a non-solvable linear equation with 2 variables. Smarandachely denied axiom, Smarandache system, non-solvable linear equa- tions, V-solution, A-solution.

Introduction Finding the exact solution of equation system is a main but a difficult objective unless the case of linear equations in classical mathematics. Contrary to this fact, what is about the non-solvable case? In fact, such an equation system is nothing but a contradictory system, and characterized only by non-solvable equations for conclusion. But our world is overlap and hybrid. The number of non-solvable equations is more than that of the solvable.

The main purpose of this paper is to characterize the behavior of such linear equation systems. Let R m , R m be Euclidean spaces with dimensional? Now let Ti,T 2 ,-.

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An 1 Received March 6, Accepted June 5, By definition, we are easily knowing that the A-solution is nothing but the same as the classical solution. Thus, such a Smarandache system is a contradictory system. Generally, we know the conception of Smarandache multi-space with its underlying combinatorial structure defined following.

## International Journal of Mathematical Combinatorics, Vol.2, | Linfan Mao - acefanodewud.cf

We consider the simplest case, i. For terminologies and notations not mentioned here, we follow -  for linear algebra,  and  for graphs and topology. A Necessary and Sufficient Condition for Non-Solvable Linear Equations The following result on non-solvable linear equations is well-known in linear algebra -.

We introduce the conception of parallel linear equations following. Thus the matrix A! Such a transformation naturally induces a transformation of linear equation system LEq , denoted by T LEq.

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By applying Theorem 2. Conseciuently, the linear equation system LEq is also non-solvable. Contradicts to the assumption. Let Lj be the tth linear equation.

Then the following conclusion is clear by definition. Tls , we can not claim that LEq is non-solvable or not.

Topics in detail to be covered are: We have proved the similiar result in . It is observed that the v-curvature tensor Shijk of M , L is the Riemannian Christoffel curvature tensor of the Riemannian space M n , gx at a point x. In the theory of curves in Euclidean space, one of the important and interesting problem is characterizations of a regular curve. But it is sufficient that an edge incident at a is a symmetric n-tuple.

We will find necessary and sufficient conditions for linear equation systems with two or three variables just by their combinatorial structures in the following sections. We get the following result. This completes the proof. Such an intersection-free graph is clearly a planar graph with each edge a straight segment since all intersection of edges appear at vertices.

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For example, let the linear equation system be LEq2 with " 1 1 3 1 2 3 1 2 4 Then its intersection-free graph I[LEq2] is shown in Fig. The following result characterizes the combinatorial structure of non-solvable linear equa- tion systems with two variables by intersection-free graphs I[LEq2], Theorem 3. Conversely, let H be a planar graph with each edge a straight segment on the plane R 2. To get the free app, enter your mobile phone number.

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