where Z(x) is some properly chosen vector of

monomials.

(3) For

monomials [m.sub.1], [m.sub.2] [member of] R and coefficients [c.sub.1], [c.sub.2] [member of] A, the least common multiple of two terms [c.sub.1][m.sub.1][e.sub.i], [c.sub.2][m.sub.2][e.sub.i] [member of] F is given by lcm ([c.sub.1][m.sub.1][e.sub.i], [c.sub.2][m.sub.2][e.sub.j]

Yang and Cao [20] proposed a

monomial geometric programming subject to max-min fuzzy relation equations with the objective function being [mathematical expression not reproducible].

Here sparse means that the degree of the polynomial is 1 and there are at most m terms of

monomials that are nonzero.

where [C.sub.0](v) is the sum of all the

monomials that do not contain u and [C.sub.1] (u, v) is the result of dividing all u-containing

monomials by u.

Let us start with the explicit expression which is obtained from expanding the exponential and differentiating the

monomial, i.e.,

Let d be a

monomial from the second subset and assume that the indices of its coefficients form a permutation as a product of r disjoint cycles, by the proof of the theorem 3.1 of [8], there exist another [2.sup.p] - 1

monomials, where p = r - [rho] and [rho] is the number of disjoint cycles of length 1 and 2, such that the sum of these [2.sup.p] - 1

monomials and d is given by

with [X.sub.1] := [g.sub.n] so that all the

monomials [G.sub.n.sup.l], l = 1,...,k, can be computed in O([b.sup.3.sub.n] + [d.sub.n-1][b.sup.2.sub.n] + [d.sup.2.sub.n-1]pb.sub.n]).

Substituting S(r) with the Zth-degree

monomial [r.sup.l], we obtain

The expressions found in Sections 3 and 4 that describe the action on

monomials of the Toeplitz operators with pseudo-homogeneous symbols (both single and multisphere) are all given as integrals on the Delzant polytopes of projective spaces.

First, an introduction to Hermite polynomial tensors,

monomials and their properties is presented in Section 2 using a specific notation.

(3) The N coefficients [C.sub.a,b,c] of the FOD tensor are then estimated simply by multiplying the matrix C, of size N x M containing the

monomials of the rank-1 symmetric fourth order tensor formed from vectors [c.sub.j], by the resultant vector l of length M.