For such kind of study, researchers defined new algebraic structures whose elements are generated by elements in classical

algebraic set and indeterminate of the real world problem with respect to algebraic operations on the well defined Neutrosophic elements.

gives rise to an

algebraic set of n-1 equations with n unknowns: [p.sub.1], [p.sub.2], [p.sub.3],....

The proof of Corollary 1 shows the existence of an algebraic set [S.sup.[infinity]] [subset] [partial derivative]D such that f extends holomorphically to a neighborhood of any point from [partial derivative]D\[S.sup.[infinity]] and for all t [member of] [S.sup.[infinity]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

a) First, we prove that the mapping f is algebraic (i.e., the graph of the mapping is contained in an irreducible complex n-dimensional algebraic set in [C.sup.n] x [C.sup.n]).

where [A.sub.4] is a closed, real-algebraic set of dimension at most 2n - 4 and [A.sub.2] [union] [A.sub.3] [union] [A.sub.4] is also a closed, real algebraic set of dimension at most 2n - 3.

1) -i) Since W is an irreducible algebraic set in D of dimension n - 1, there exists an irreducible polynomial h in [C.sup.n] such that W = {z [member of] D : h(z) = 0}.

The algebraic set [A.sub.2] contains finitely many components, which we will denote as [[sigma].sub.1], ..., [[sigma].sub.N].

According to authors Vasantha Kandasamy and Smarandache, the Neutrosophic set is a nice composition of an

algebraic set and indeterminate element of the real world problem.

From now on we are interested in Seshadri curves ([C.sub.t]; ([P.sub.1]) t, ..., [([P.sub.r]).sub.t]) for L moving in a non-trivial family over some

algebraic set [DELTA].

The other six contributions examine the relationship between nonnegativity and sums of squares, compare the different notions of duality, find spectrahedral approximations of convex hulls of

algebraic sets, extend algebraic certificates of real algebraic geometry to noncommutative polynomials, and derive sums of Hermitian squares.

Appendices cover such background material as the Tarksi-Seidenberg theorem and

algebraic sets. Designed for students at the beginning graduate level, this concentrates on concrete objects, such as polynomials in n variables with real coefficients, and Marshall includes plenty of examples and new, simple proofs.

Based on a June 2005 summer school, this series of 24 lectures introduces the

algebraic sets, sheaf theory, and homological algebra leading to the definition and alternative characterizations of local cohomology.