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Decision Problem

[Graphics:../Images/index_gr_40.gif] [Graphics:../Images/index_gr_41.gif]
[Graphics:../Images/index_gr_42.gif] [Graphics:../Images/index_gr_43.gif]
[Graphics:../Images/index_gr_44.gif] [Graphics:../Images/index_gr_45.gif]
[Graphics:../Images/index_gr_46.gif] [Graphics:../Images/index_gr_47.gif]

Figure 1. Functions for deciding the existence of solutions.

Developer` and Experimental` contexts are put on $ContextPath so that we do not need to prepend them to the function names.

[Graphics:../Images/index_gr_48.gif]

This shows that [Graphics:../Images/index_gr_49.gif] is nonnegative for all real x and y.

[Graphics:../Images/index_gr_50.gif]
[Graphics:../Images/index_gr_51.gif]

Let us define two classes of inequalities.

[Graphics:../Images/index_gr_52.gif]

This shows that i1[2/3] and i2[2/3] have no common solutions.

[Graphics:../Images/index_gr_53.gif]
[Graphics:../Images/index_gr_54.gif]

Here is a graphical representation of the problem. Solutions of i1 are colored blue and solutions of i2 are colored red.

[Graphics:../Images/index_gr_55.gif]

[Graphics:../Images/index_gr_56.gif]

This shows that i1[1/3] and i2[1/3] have common solutions.

[Graphics:../Images/index_gr_57.gif]
[Graphics:../Images/index_gr_58.gif]

Here is a graphical representation of the problem. Solutions of i1 are colored blue and solutions of i2 are colored red.

[Graphics:../Images/index_gr_59.gif]

[Graphics:../Images/index_gr_60.gif]

The next example is based on a problem formulated by T. Ono in 1914. He conjectured that for all triangles the following inequality is true.

[Graphics:../Images/index_gr_61.gif]

The variables a, b, and c are lengths of sides of the triangle, and F is its area. A counterexample was found by G. Quijano in 1915. We can generate a counterexample using InequalityInstance.

[Graphics:../Images/index_gr_62.gif]
[Graphics:../Images/index_gr_63.gif]

For acute triangles the conjecture is true, and the proof was given by F. Balitrand in 1916. Here is a proof using ImpliesRealQ.

[Graphics:../Images/index_gr_64.gif]
[Graphics:../Images/index_gr_65.gif]

Converted by Mathematica      April 24, 2000

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