## Examples of *P0* in a sentence

If the currency in which the Contract Price

**P0**is expressed is different from the currency of origin of the labor and material indices, a correction factor will be applied to avoid incorrect adjustments of the Contract Price.Hence, the membership of (

**P0**+ P1)2 − (**P0**+ P1) in Ψ−2,−2 is equivalent to the principal symbol p1 of P1 satisfyinge1 + Πp1 + p1Π − p1 = 0.We compute this:01(

**P0**+ P1)2 − (**P0**+ P1) = P 2 −**P0**+ P0P1 + P1P0 − P1 + P 2 = E1 + P0P1 + P1P0 − P1 + F2,scscwhere F2 ∈ Ψ−2,−2, so irrelevant for our conclusion on the improved projection property.We first write down the argument with U1 = U = ∂scT∗X, i.e. globally, and then simply remark on its microlocal nature.sc20One starts by taking any operator

**P0**∈ Ψ0,0 with principal symbol Π; one can replace**P0**by 1 (**P0**+ P∗)0scand thus assume that it is self-adjoint.It is observed that as the flotation time increases (from

**P0**to P4) the froth height decreases (Figure 3a).**P0**is obtained by solving the above ME with the initial condition PM (t = 0) = δMM0 :00P (t|M ) = h1 − e−p, t µ(t')dt' iM0 .Aθ pP f /¸ BThere is a map θ0 :

**P0**→ A0 which lifts f0:A0θ0 p0P0 f0B¸/ 0Suppose given a lift up to degree n, ie.For this, we project equation (2.1a) onto the kernel of the constraint which gives u˙[1] − P0A(u[1] + u[2]) = P0F.Here we have used P0D−λ = 0 which follows from D− : Q → Xc. Thus, we obtain withA0 = P0A|X0 ,u˙[1] − A0u[1] =

**P0**Au[2] + Ffor which we can use the variation-of-constants formula for evolution equations, since A0 is assumed to generate an analytic semigroup.Moreover, the composition of f with the exact solution satisfies f (u) ∈ D(A).(A2) The initial condition is consistent, i.e., Du0 = G(0) and u0 ∈ D(A).D → D(A3) The constraint operator : X Q is linear, onto, and its kernel X0 := ker is a closed subspace of X.(A4) There exists a right-inverse D− : Q → X such that DD−q = q for all q ∈ Q.The existence of a right-inverse implies that X0 is a complemented subspace with pro- jection

**P0**:= id −D−D : X → X0 and complement Xc := ker**P0**= im D−.The above result stems from the second version of the density:1 Z 1 R 1 2 2= ,ƒWN (λ) 2ve— 2H0 (h+p ) A(p) dP0(p),where

**P0**is the CBM conditioned to be mean-zero.