Second-order data

The values used in plotting Figs. 2-1 and 2-2 can be used to illustrate the method for first-order and second-order data. Plots of t/E versus time are shown in Fig. 2-9. The second-order data define a precise straight line, and those for n = 1 are linear to E < 0.4. The latter graph has a slope of 0.6, giving n = 1.2. 

Wilkinson s method for the estimation of the reaction order is illustrated for first-order (left) and second-order (right) kinetic data. The first-order reaction is the decomposition of diacetone alcohol (Table 2-3 and Fig. 2-4) data to about 50 percent reaction are displayed. The slope gives an approximate order of 1.2. The second-order data (Fig. 2-2) give a precise fit to Eq. (2-59) and an order of two exactly. 

Figure 75-1 (a) Second order data (amplitude, multiple frequencies, time) (b) Second order data (amplitude at one frequency, with X and Y spatial dimensions). 

In [Le Barz (10)] these varieties are shown to be smooth for X a smooth variety over C. The Ei X) are irreducible divisors in H3(X). D (X) is the variety of second order data on X, which we want to study in more detail in chapter 3. 

In section 3.2 we consider the varieties of higher order data D X). Their definition is a generalisation of that of D X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D X) is a natural desingularisation of. Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C Pn with linear subspaces of P. ... 

Let X be a smooth projective variety of dimension d over an algebraically closed field k. In this section we want to define a variety D (X) of second order data of m-dimensional subvarieties of X for any non-negative integer m < d. A general point of D ln X) will correspond to the second order datum of the germ of a smooth m-dimensional subvariety Y C X in a point x X, i.e. to the quotient of Ox,x- Assume for the moment that the ground field is C and x Y C X, X is a smooth complex d-manifold and we have local coordinates zi,..., at x. Then Y is given by equations... 

Later we will see that D X) is reduced and even smooth. D2n(X) is called the variety of second order data of m-dimensional subvarieties of X. Analogously we define D n(X) as the closed subscheme of X x Xtm+1l that represents the functor given by... 

For a surface 5 the variety D2(S) is considered in the literature (using a slightly different definition). It is called the variety of second order data on S and denoted... 

D iX) as the variety of second order data of m-dimensional subvarieties of X. 

We want to see in what respect D X) parametrizes the second order data of m-dimensional subvarieties of X. First we will more generally consider the Ith order data of germs of smooth subvarieties. 

Now we want to compute the class of the complement D (X)<, = D (X) D (X)0. It parametrizes in a suitable sense the second order data of singular m-dimensional subvarieties of X. We will use a tool that will play a major role in the enumerative applications of higher order data in section 3.2, the Porteous formula. We will not quote the result in full generality but in the formulation in which we are going to use it. 

Tauler, R. (1995), Multivariate curve resolution applied to second order data, Chemomet. Intell. Lab. Syst., 30,133-146. 

While beyond the scope of this chapter, A-way modeling methods are being used more widely in the literature [65], The idea here is to use other dimensions of information. For example, first-order data consists of only the spectroscopic order for a spectrum or the chromatographic order for a chromatogram. Second-order data is that formed by combing data from two first-order instruments. Variance expressions for N-way modeling have been derived [66, 67], See Chapter 12 for more information. 

A standard addition method has been studied for use with second-order data [87], The specific application investigated was analysis of trichloroethylene in samples that have matrix effects caused by an interaction with chloroform. 

---