Crescent Singularities in Crumpled Sheets

We examine the crescent singularity of a developable cone in a setting similar to that studied by Cerda et al [Nature 401, 46 (1999)]. Stretching is localized in a core region near the pushing tip and bending dominates the outer region. Two types of stresses in the outer region are identified and shown to scale differently with the distance to the tip. Energies of the d-cone are estimated and the conditions for the scaling of core region size R_c are discussed. Tests of the pushing force equation and direct geometrical measurements provide numerical evidence that core size scales as R_c ~ h^{1/3} R^{2/3}, where h is the thickness of sheet and R is the supporting container radius, in agreement with the proposition of Cerda et al. We give arguments that this observed scaling law should not represent the asymptotic behavior. Other properties are also studied and tested numerically, consistent with our analysis.Comment: 13 pages with 8 figures, revtex. To appear in PR

Early career casual teachers: The role of relationships with colleagues in negotiating a teacher identity and developing resilience

Developing relationships with colleagues has been identified as one way to enhance teacher resilience and assists in negotiating a professional identity. For early career teachers, opportunities to participate in induction and mentoring programmes and engage in professional learning can assist in developing these relationships. However, for early career teachers who can only obtain casual work and work intermittently often in many different schools, these opportunities may be limited. This chapter presents longitudinal, qualitative research that explores how early career casual teachers negotiated their teacher identity. Drawing on data from focus groups, semi-structured interviews and reflective tasks, the chapter shares insights into how relationships are pivotal in the development of a strong teacher identity

Spontaneous curvature cancellation in forced thin sheets

In this paper we report numerically observed spontaneous vanishing of mean curvature on a developable cone made by pushing a thin elastic sheet into a circular container. We show that this feature is independent of thickness of the sheet, the supporting radius and the amount of deflection. Several variants of developable cone are studied to examine the necessary conditions that lead to the vanishing of mean curvature. It is found that the presence of appropriate amount of radial stress is necessary. The developable cone geometry somehow produces the right amount of radial stress to induce just enough radial curvature to cancel the conical azimuthal curvature. In addition, the circular symmetry of supporting container edge plays an important role. With an elliptical supporting edge, the radial curvature overcompensates the azimuthal curvature near the minor axis and undercompensates near the major axis. Our numerical finding is verified by a crude experiment using a reflective plastic sheet. We expect this finding to have broad importance in describing the general geometrical properties of forced crumpling of thin sheets.Comment: 13 pages, 12 figures, revtex

A simplicial gauge theory

We provide an action for gauge theories discretized on simplicial meshes, inspired by finite element methods. The action is discretely gauge invariant and we give a proof of consistency. A discrete Noether's theorem that can be applied to our setting, is also proved.Comment: 24 pages. v2: New version includes a longer introduction and a discrete Noether's theorem. v3: Section 4 on Noether's theorem has been expanded with Proposition 8, section 2 has been expanded with a paragraph on standard LGT. v4: Thorough revision with new introduction and more background materia

Measurement of the spatial dependence of temperature and gas and soot concentrations within large open hydrocarbon fuel fires

A series of large-scale JP-4 fuel pool fire tests was conducted to refine existing mathematical models of large fires. Seven tests were conducted to make chemical concentration and temperature measurements in 7.5 and 15 meter-diameter pool fires. Measurements were made at heights of 0.7, 1.4, 2.9, 5.7, 11.4, and 21.3 meters above the fires. Temperatures were measured at up to 50 locations each second during the fires. Chemistry samples were taken at up to 23 locations within the fires and analyzed for combustion chemistry and soot concentration. Temperature and combustion chemistry profiles obtained during two 7.5 meter-diameter and two 15 meter-diameter fires are included

Rim curvature anomaly in thin conical sheets revisited

This paper revisits one of the puzzling behaviors in a developable cone (d-cone), the shape obtained by pushing a thin sheet into a circular container of radius $R$ by a distance $\eta$ [E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999)]. The mean curvature was reported to vanish at the rim where the d-cone is supported [T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006)]. We investigate the ratio of the two principal curvatures versus sheet thickness $h$ over a wider dynamic range than was used previously, holding $R$ and $\eta$ fixed. Instead of tending towards 1 as suggested by previous work, the ratio scales as $(h/R)^{1/3}$. Thus the mean curvature does not vanish for very thin sheets as previously claimed. Moreover, we find that the normalized rim profile of radial curvature in a d-cone is identical to that in a "c-cone" which is made by pushing a regular cone into a circular container. In both c-cones and d-cones, the ratio of the principal curvatures at the rim scales as $(R/h)^{5/2}F/(YR^{2})$, where $F$ is the pushing force and $Y$ is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.Comment: 25 pages, 12 figures. Added references. Corrected typos. Results unchange

The Casimir force on a surface with shallow nanoscale corrugations: Geometry and finite conductivity effects

We measure the Casimir force between a gold sphere and a silicon plate with nanoscale, rectangular corrugations with depth comparable to the separation between the surfaces. In the proximity force approximation (PFA), both the top and bottom surfaces of the corrugations contribute to the force, leading to a distance dependence that is distinct from a flat surface. The measured Casimir force is found to deviate from the PFA by up to 15%, in good agreement with calculations based on scattering theory that includes both geometry effects and the optical properties of the material

Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup S as follows: Players 1 and 2 choose elements x and y in S, respectively, and player 1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.Comment: 17 pages. To appear in International Journal of Game Theor

Light-Cone Quantization of the Liouville Model

We present the quantization of the Liouville model defined in light-cone coordinates in (1,1) signature space. We take advantage of the representation of the Liouville field by the free field of the Backl\"{u}nd transformation and adapt the approch by Braaten, Curtright and Thorn. Quantum operators of the Liouville field $\partial_{+}\phi$, $\partial_{-}\phi$, $e^{g\phi}$, $e^{2g\phi}$ are constructed consistently in terms of the free field. The Liouville model field theory space is found to be restricted to the sector with field momentum $P_{+}=-P_{-}$, $P_{+}> 0$ , which is a closed subspace for the Liouville theory operator algebra.Comment: 16 p, EFI-92-6