Equality linear system, residuals

In a linear polyesterification involving bifunctional monomers exclusively, the hydroxyl and carboxyl groups being present in equal numbers, the number of unreacted carboxyl groups must equal the number of molecules present in the system, provided that no side reactions occur the number of molecules in moles per unit volume is equal, therefore, to co(l—p). If each residue from a glycol and from a di-... 

Thus, the linear response function has poles where the absolute value of the frequency is equal to an excitation energy of tire system. From the corresponding residues... 

Scheme 3. The Lipid Cycle in the Biosynthesis of Linear Peptidoglycan in S. auretM. [Although the amidation reaction (111) is shown here as though it precedes the addition of pentaglycine peptide (IV), there is no evidence as to the sequence in which these reactions occur. The amidation reaction proceeds equally well before the addition of 2-acetamido-2-deoxy-D-glucose residues (reaction II), at least in a cell-free system. ]...

In Section 7.3, we showed how to automatically calculate the weights for the equations of a system such that the residual is equal to the distance of the working point from the hyperplane of the linearized equation. 

The weight function is identical in the Galerkin and tau methods (i.e. weight function is set equal to the basis function in the polynomial trial function expansion, see (12.405)-(12.408)). The essential difference between the Galerkin and the tau methods is the treatment of the boundary conditions (strong form). In the tau method, the boundary conditions are enforced as additional equations and in order to get a system of equations where the number of unknowns is identical to the number of equations, the equation system has to be relaxed such that n residual equations are replaced by the n boundary conditions. In the Galerkin method, some linear combinations of the polynomials that fulfill the boundary conditions must be performed. 

---