Synthetic research on the relationship among various equivalence relations of singularities

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T.C.Kuo introduced the notion of blow-analytic equivalence as a desirable equivalence relation for real analytic singularities. In this research, we have proved that C^1 equivalence implies blow-analytic one. This is a negative result to the conjecture given by Kuo and other many real singularitists. In addition, we have proved that the directional dimension of definable sets in o-minimal structure over general real closed field is a Lipschitz invariant, generalizing the Lipschitz invariant I introduced in order to show that blow-analytic equivalence does not always imply Lipschitz one.