Factor [k.sub.1] together with [k.sub.2] make an

affine function of maximal stock level depending on market demand, given in the following form

Under this structure, Duffie and Kan (1996) show that the yield for any maturity [tau] can be obtained as an

affine function of the state vector, that is,

By definition, the piecewise

affine function [omega] of a non-centered Young diagram is given by x [??] [[omega].sub.[lambda]](x - c) for some integer c and usual Young diagram [lambda]--see the right- most part on Figure 1.

where [[??].sup.*.sub.i] [member of] [R.sup.Nm+n] denotes the piecewise

affine function of [[??].sup.*] in partition [P.sub.i], [[GAMMA].sub.i] [member of] [R.sup.(Nm+n)xn], i = 1, ..., r, and [[eta].sub.i] [member of] [R.sup.Nm+n], i = 1, ..., r.

A nonconstant [GAMMA]-linear

affine function cannot have a horizontal point of inflection [x.sub.0] [member of] (a,b).

--Dependency: First, note that the dependency (z [right arrow] f(z)), by itself, simply denotes the

affine function, f.

In the case where the function h is an

affine function, we can obtain more definite result about total effort level: For [Sigma] [is less than or equal to] 2, total effort level is always increasing in the valuation parameter [Alpha]; for [Sigma] [is greater than] 2, total effort level is increasing in [Alpha] until [Alpha] reaches 1/([Sigma] - 2) and then is decreasing in [Alpha].

Definition 4

Affine Function: A Boolean function which can be expressed as 'xor' ([direct sum]) of some or all of its input variables and a Boolean constant is an

affine function.

Here, A is the affine structure as a [Z.sup.dim(B)]-local system in the tangent bundle T(B [DELTA]), [-.sup.[disjunction]] denotes -'s dual local system, [[??].sup.[disjunction]] is the local system of

affine functions.

Formula (19) defines a system of equations giving univocally [[tau].sup.*.sub.e] and [[bar.[tau]].sup.*.sub.e] as fractions (ratios) of two

affine functions in [mathematical expression not reproducible] and [mathematical expression not reproducible].

Wasowicz, "A sandwich theorem and HyersUlam stability of

affine functions," Aequationes Mathematicae, vol.

Moreover, the profit functions of the supply chain members are all

affine functions of the whole supply chain's profit function.

From (6), it is shown that the profit functions of the supply chain members are all

affine functions of the whole supply chain's profit function.

It is clear that the multiplication, the division, the square, and the square root are neither affine operations nor

affine functions. Therefore, the results have not an affine form, and thus an affine approximation must be introduced.