Differential index

From equation (1.5.7), change in the concentration of an element i with the differentiation index (FeO/MgO)liq, written in lieu of (CFe0/CMg0)Hq, is calculated as... 

Differential index indicates statistical significance of differences (P < 0.05 —0.001). 

For purposes of analysis, this IDN can be regarded as a fixed-coupon bond plus the double indexation of an interest rate differential. Indexation here refers to a reference rate based on an index. The double indexation creates two long positions in a 5-year dollar-denominated fixed-rate note and two short positions in a euro-denominated fixed-rate note. The short positions remove the euro exchange-rate risk, so investors are exposed only to the euro interest rate risk, which is the desired exposure. 

A characteristic of DAEs besides their form is their differentiation index [32]. For a definition and an example see Appendix C. It is an indicator for the problems to be encountered with the numerical solution of a set of DAEs. Systems of index > 1 are usually called higher index DAEs and the higher the index the more severe numerical difficulties can be. As the mathematical description of problems in various disciplines often leads to DAE system, they have been a research subject for more than two decades. A large body of publications and a number software programs for their numerical solution have emerged. DAE systems of index 1 can be safely numerically computed by means of the backward differentiation formula (BDF) [33, 34] implemented in solvers such as the well known and widely used DASSL code [35]. 

If the matrix A22 A11 is non-singular, (2.24) differentiated with respect to time can be solved for dX/dt. According to the definition of the differentiation index, the DAE in this system mode is of index 2. 

Then (C. 1) has a differentiation index v if v is the minimal number of analytical differentiations such that analytical manipulations of the set of equations... 

In this subsection, we shall deal with (possibly, infinite-dimensional) Lie algebras of functions on a manifold M with the Poisson bracket, we shall define in terms of the Lie algebra of (first) integrals the differential index of such Lie algebras and shall give the integrabUity condition in quadratures of dynamic systems on M Hamiltonian with respect to the Poisson bracket, ... 

From an appropriate process modelling it follows that the system (1.1) has the differential index 1. Usually it is a stiff system, whose discretization and linearization yields systems of equations with sparse nonsymmetric Jacobian matrices. The systems can comprise several 10 000 equations and are hierarchically structured into subsystems in accordance with the functional units of the chemical plant ... 

Minimality for nonlinear state-space models Lumped process models with differential index 1 can be described by input-affine state-space models, where the notion of minimality is related to the number of state variables necessary and sufficient to produce a given input-output behaviour. This means, that one naturally uses the single quality index = dim x = n for both linear and nonlinear state-space models and orders the functionally equivalent models accordingly. 

The operating point(s) [x z ] can be determined for prescribed input values u by solving (1-2) with x = 0 which means the solution of an algebraic system of equations. A necessary condition on the solvability of the system of equations above is that the number of differential (algebraic) equations equals to the number of differential (algebraic) variables (degree of freedom equals to zero), and tiie original DAE system has differential index 1. 

Soares and Secchi (2002) have proposed a structural method for index reduction and solvability test of DAE systems. With this method, structural singularity can be tested and the structural differential index can be reduced to zero by adding new variables and equations. Such variables and equations are the derivatives of the original ones with respect to the independent variable. 

There is another way to formally determine the index of a DAE, sometimes called the differentiation index, as the index is determined by successively differentiating the constraints ... 

The differentiation index is defined as the number of differentiations required to obtain an ODE in all unknowns. Thus, the equations of motion with the constraint given on position level are an index-3 system conformal to their perturbation index. Analogously, the velocity constrained equations are an index-2 problem and the acceleration constrained equations an index-1 system. 

We consider in this book exclusively DAEs of the form (5.1.2) with a regular matrix Ei y). For these systems the differentiation index and the perturbation index are the same, [HLR89]. 

---