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Abstract: Rayleigh functionals are the generalization of Rayleigh quotients for matrices to nonlinear eigenvalue problems. While analyzing the state of the art we address some problems concerning the existing definitions and give different new ones. These new definitions are more general in that they allow complex eigenvalues and operators. However, in order to obtain local existence and uniqueness, we need to restrict to target eigentriplets with right and left normalized eigenvectors such that and to sufficiently good eigenvector approximations defining the Rayleigh functional. As there is a one-sided and a two-sided Rayleigh quotient, we introduce the one-sided and the two-sided Rayleigh functional. We estimate the quality of the Rayleigh functional in terms of the angles between target eigenvectors and their approximations and derive perturbation bounds and first order perturbation results of the same kind and order as known for linear problems. We show that stationarity holds in the same way. We conclude with an analysis of the so called generalized Rayleigh quotient proposed by Lancaster for matrix polynomials. We study the connection between this generalized Rayleigh quotient and the two-sided Rayleigh functional, and compare another variant without left eigenvector approximations with the one-sided Rayleigh functional. [Copyright &y& Elsevier]

Late infantile neuronal ceroid lipofuscinosis, a fatal neurodegenerative disease of childhood, is caused by mutations in the TPP1 gene that encodes tripeptidyl-peptidase I. We show that purified TPP1 requires at least partial glycosylation for in vitro autoprocessing and proteolytic activity. We crystallized the fully glycosylated TPP1 precursor under conditions that implied partial autocatalytic cleavage between the prosegment and the catalytic domain. X-ray crystallographic analysis at 2.35 Å resolution reveals a globular structure with a subtilisin-like fold, a Ser475-G1u272-Asp360 catalytic triad, and an octahedrally coordinated Ca2+-binding site that are characteristic features of the S53 sedolisin family of peptidases. In contrast to other S53 peptidases, the TPP1 structure revealed steric constraints on the P4 substrate pocket explaining its preferential cleavage of tripeptides from the unsubstituted N terminus of proteins. Two alternative conformations of the catalytic Asp276 are associated with the activation status of TPP1. 28 disease-causing missense mutations are analyzed in the light of the TPP1 structure providing insight into the molecular basis of late infantile neuronal ceroid lipofuscinosis. [ABSTRACT FROM AUTHOR]