Closed but provocative questions: curves enclosing unit area

This article describes a task leading to work on curve sketching, simultaneous equations and integration to find the area enclosed between two curves. An initial closed question is used to confront students with a provocative answer, which they then explore in a much more open-ended way

Lower Bound for Convex Hull Area and Universal Cover Problems

In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a unit length curve has area at least 0.232239. In addition, we show that a convex universal cover for a unit closed curve has area at least 0.0879873.Comment: 12 pages, 9 figure

CII* Absorption in Damped Lyman Alpha Systems: (II) A New Window on the Star Formation History of the Universe

Starting from the SFR per unit physical area, determined for DLAs using the C II* method, we obtain the SFR per unit comoving volume at $z$ $\approx$ 3, and find that it agrees with that for the Lyman Break Galaxies (LBGs). Though the mass of produced stars indicated by the SFRs is consistent with the current densities of known stellar populations, the mass of metals produced by $z$=2.5 is 30 times larger than detected in absorption in DLAs. The most likely solutions to this ``missing metals'' problem is that star formation occurs in compact bulge regions. We search for evidence of feedback and find no correlations between the SFR per unit area and N(H I), but possible correlations between SFR per unit area and low-ion velocity width and SFR per unit area and metal abundance. We show that (a) the correlation between cooling rate and dust-to-gas ratio is positive evidence for grain photoelectric heating, (b) the CMB does not significantly populate the C II excited fine-structure states, and (c) the ratio of CII* to resonance-line optical depths is a sensitive probe of the multi-phase structure of the DLA gas. We address recent arguments that DLAs are comprised only of WNM gas, and show them to be inconclusive. Despite the rough agreement between SFR per unit comoving volume for DLAs and LBGs, current evidence indicates these are distinct populations

Statistical mechanical properties of polymer configurations which enclose a constant area

The statistical mechanical properties of plane polymer loops enclosing a constant area are investigated, using a continuous model from the start. For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit

Plane polymer configurations enclosing a fixed area in an electric field: generating functional and statistical mechanical properties

The statistical mechanical properties of plane polymer configurations which enclose a fixed area and are subject to an external electric field are investigated. For this purpose an exact expression for the generating functional is obtained and subsequently used to derive: (a) the distribution function for the enclosed area; (b) the mean square distance of a given repeating unit from the origin; (c) the entropic force on a repeating unit