Introduction to Principal Components Analysis

Understanding the inverse equivalent width - luminosity relationship (Baldwin Effect), the topic of this meeting, requires extracting information on continuum and emission line parameters from samples of AGN. We wish to discover whether, and how, different subsets of measured parameters may correlate with each other. This general problem is the domain of Principal Components Analysis (PCA). We discuss the purpose, principles, and the interpretation of PCA, using some examples from QSO spectroscopy. The hope is that identification of relationships among subsets of correlated variables may lead to new physical insight.Comment: Invited review to appear in ``Quasars and Cosmology'', A.S.P. Conference Series 1999. eds. G. J. Ferland, J. A. Baldwin, (San Francisco: ASP). 10 pages, 2 figure

A tale of two airfoils: resolvent-based modelling of an oscillator vs. an amplifier from an experimental mean

The flows around a NACA 0018 airfoil at a Reynolds number of 10250 and angles of attack of alpha = 0 (A0) and alpha = 10 (A10) are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity profiles are data-assimilated so that they are solutions of the incompressible Reynolds-averaged Navier-Stokes equations forced by Reynolds stress terms which are derived from experimental data. Spectral proper orthogonal decompositions (SPOD) of the velocity fluctuations and nonlinear forcing find low-rank behaviour at the shedding frequency and its higher harmonics for the A0 case. In the A10 case, low-rank behaviour is observed for the velocity fluctuations in two bands of frequencies. Resolvent analysis of the data-assimilated means identifies low-rank behaviour only in the vicinity of the shedding frequency for A0 and none of its harmonics. The resolvent operator for the A10 case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by SPOD. It is also shown that the second linear mechanism, corresponding to the Kelvin-Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as a mean flow due to the fact that experimental data are missing near the leading edge. The A0 case is classified as an oscillator where the flow is organised around an intrinsic instability while the A10 case behaves like an amplifier whose forcing is unstructured. For both cases, resolvent modes resemble those from SPOD when the operator is low-rank. To model the higher harmonics where this is not the case, we add parasitic resolvent modes, as opposed to classical resolvent modes which are the most amplified, by approximating the nonlinear forcing from limited triadic interactions of known resolvent modes.Comment: 32 pages, 23 figure

Phenomenon of ordinariness in nursing

This phenomenological research aimed to illuminate the nature and effects of ordinariness in nursing and to discover whether the phenomenon enhanced the nursing encounter. The researcher worked as a participant observer with six registered nurses in a Professorial Nursing Unit. Following each interaction, the researcher wrote her impressions in a personal-professional journal and audiotaped conversations with the respective nurses and patients to gain their impressions. Using a theoretical framework of the phenomenological concepts of lived experience, Dasein, Being-in-the-world and fusion of horizons as an underpinning methodology, an initial hermeneutical analysis and interpretation of the impressions generated qualities and activities indicative of the aspects of the phenomenon of ordinariness in nursing. The second phase of the analysis and interpretation sought to illuminate the nature of the phenomenon itself. Eight actualities of the nature of the phenomenon emerged: \u27allowingness,\u27 \u27straightforwardness,\u27 \u27self-likeness,\u27 \u27homeliness,\u27 \u27favourableness,\u27 \u27intuneness,\u27 \u27lightheartedness\u27 and \u27connectedness.\u27 These actualities were described in relation to the phenomenon of interest. The effects of the phenomenon were the creative potential to enhance the nursing encounter and included many and various effects of facilitation, fair play, familiarity, family, favouring, feelings, fun and friendship. The research found that nurses and patients shared a common sense of humanity, which enhanced the nursing encounter. Within the context of caring, the nurses were ordinary people, perceived as being extraordinarily effective, by the very ways in which their humanness shone through their knowledge and skills, to make their whole being with patients something more than just professional helping. The shared sense of ordinariness between nurses and patients made them as one in then- humanness and created a special place, in which the relative strangeness of the experience of being in a health care setting, could be made familiar and manageable

On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

The relationship between Koopman mode decomposition, resolvent mode decomposition and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalised to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatio-temporal Koopman operator, which has a travelling wave interpretation. The relationship leads to a generalisation of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known