FRACTAL STOKES’ THEOREM BASED ON INTEGRALS ON FRACTAL MANIFOLDS

In this paper, with the Hausdorff measure, the Hausdorff integral on fractal sets with one or lower dimension is firstly introduced via measure theory. Then the definition of the integral on fractal sets in [Formula: see text] is given. With the variable substitution theorem in the Riemann integral generalized to the integral on fractal sets, the integral on fractal manifolds is defined. As a result, with the generalization of Gauss’ theorem, Stokes’ theorem is generalized to the integral on fractal manifolds in [Formula: see text].

ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS

Let$R\subset F$be an extension of real closed fields, and let${\mathcal{S}}(M,R)$be the ring of (continuous) semialgebraic functions on a semialgebraic set$M\subset R^{n}$. We prove that every$R$-homomorphism${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$is essentially the evaluation homomorphism at a certain point$p\in F^{n}$adjacent to the extended semialgebraic set$M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when$M$is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

Term Substitution Theorem in First-Order Fuzzy Predicate System

It has given out term substitution theorem in the system K *. R0-algebra is introduced firstly, and then the fuzzy interpretation of the first-order language. It discusses term substitution theorem in the first-order fuzzy predicate system K *.

ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

In this work we define a semialgebraic set S ⊂ ℝn to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.