A syllabus of line geometry

In the study of advanced geometry, we shall deal with a certain important relation between pairs of figures in space, and also between their properties. There are two distinct parts to analytic geometry, the analytic work and the geometric interpretation. Two systems of geometry, depending upon different elements with the same number of coordinates, will have the same analytic expressions and will differ only in the interpretation of the analysis. In such a case it is often sufficient to know the meaning of the coordination and the interpretation of a few fundamental relations in each system in order to find for a theorem in one geometry a corresponding theorem in the other. The nature of this relation is explained by the theorem of duality which assets that a dual, or reciprocal, statement can be derived from a given statement

A Detailed Look at Chemical Abundances in Magellanic Cloud Planetary Nebulae. I. The Small Magellanic Cloud

We present an analysis of elemental abundances of He, N, O, Ne, S, and Ar in Magellanic Cloud planetary nebulae (PNe), and focus initially on 14 PNe in the Small Magellanic Cloud (SMC). We derived the abundances from a combination of deep, high dispersion optical spectra, as well as mid-infrared (IR) spectra from the Spitzer Space Telescope. A detailed comparison with prior SMC PN studies shows that significant variations among authors of relative emission line flux determinations lead to systematic discrepancies in derived elemental abundances between studies that are >~0.15 dex, in spite of similar analysis methods. We used ionic abundances derived from IR emission lines, including those from ionization stages not observable in the optical, to examine the accuracy of some commonly used recipes for ionization correction factors (ICFs). These ICFs, which were developed for ions observed in the optical and ultraviolet, relate ionic abundances to total elemental abundances. We find that most of these ICFs work very well even in the limit of substantially sub-Solar metallicities, except for PNe with very high ionization. Our abundance analysis shows enhancements of He and N that are predicted from prior dredge-up processes of the progenitors on the AGB, as well as the well known correlations among O, Ne, S, and Ar that are little affected by nucleosynthesis in this mass range. We identified MG_8 as an interesting limiting case of a PN central star with a ~3.5 M_sun progenitor in which hot-bottom burning did not occur in its prior AGB evolution. We find no evidence for O depletion in the progenitor AGB stars via the O-N cycle, which is consistent with predictions for lower-mass stars. We also find low S/O ratios relative to SMC H_II regions, with a deficit comparable to what has been found for Galactic PNe.Comment: 9 figures, 6 tables; to be published in Ap

Turbulent flow in a ribbed channel: Flow structures in the vicinity of a rib

PIV measurements are performed in a channel with periodic ribs on one wall. The emphasis of this study is to investigate the flow structures in the vicinity of a rib in terms of mean velocities, Reynolds stresses, probability density functions (PDF), and two-point correlations. The PDF distribution of u0 is bimodal in the separated shear layer downstream of the rib. The maximum Reynolds shear stresses occur at the leading edge of the rib. Based on quadrant analysis, it is found that ejection motions make a dominant contribution to the Reynolds shear stress in this region. Moreover, topology-based visualization is applied to the separation bubble upstream of the rib. Salient critical points and limit cycles are extracted, which gives clues to the physical processes occurring in the flow

Respondent Consistency in a Tournament-Style Contingent Choice Survey

We present the results of an internet-based contingent choice survey about management options at North Cascades National Park, focusing on respondent consistency. A tournament-style contingent ranking design followed by a contingent rating exercise allows for tests of different kinds of consistency in survey responses. Many respondents give inconsistent responses, but these inconsistencies do not create large differences in estimated tradeoffs between scenario attributes

Space-Time Correlations and Spectra of Wall Pressure in a Turbulent Boundary Layer

Measurements of the statistical properties of the fluctuating wall pressure produced by a subsonic turbulent boundary layer are described. The measurements provide additional information about the structure of the turbulent boundary layer; they are applicable to the problems of boundary-layer induced noise inside an airplane fuselage and to the generation of waves-on water. The spectrum of the wall pressure is presented in dimensionless form. The ratio of the root-mean-square wall pressure to the free-stream dynamic pressure is found to be a constant square root of bar P(sup 2)/q(sub infinity) = 0.006 independent of Mach number and Reynolds number. In addition, space- time correlation measurements in the stream direction show that pressure fluctuations whose scale is greater than or equal to 0.3 times the boundary-layer thickness are convected with the convection speed U(sub c) = 0.82U(sub infinity) where U(infinity) is the free-stream velocity and have lost their identity in a distance approximately equal to 10 boundary-layer thicknesses