Crumpling a thin sheet

Crumpled sheets have a surprisingly large resistance to further compression. We have studied the crumpling of thin sheets of Mylar under different loading conditions. When placed under a fixed compressive force, the size of a crumpled material decreases logarithmically in time for periods up to three weeks. We also find hysteretic behavior when measuring the compression as a function of applied force. By using a pretreating protocol, we control this hysteresis and find reproducible scaling behavior for the size of the crumpled material as a function of the applied force.     

Instability of the Kaluza-Klein vacuum

It is argued that the ground state of the Kaluza-Klein unified theory is unstable against a process of semiclassical barrier penetration. This is related to the fact that the positive energy conjecture does not hold for the Kaluza-Klein theory; an explicit counter-example is given. The reasoning presented here assumes that in general relativity one should include manifolds of non-vacuum topology. It is argued that the existence of elementary fermions (not present in the original Kaluza-Klein theory) would stabilize the Kaluza-Klein vacuum.     

Chiral symmetry,the 1/N expansion and the SU(N) thirring model

In the two-dimensional SU(N) Thirring model, the 1/N expansion seems to predict spontaneous breaking of the continuous chiral symmetry. This is impossible in two-dimensions. Reasoning along the lines of Berezinski, Kosterlitz and Thouless for the two-dimensional XY model, we argue that, in fact, rather than showing long-range order, $\text{〈}\text{ψ}\text{ψ(x)}\text{ψ}\text{ψ(0)〉}$ vanishes in this model as |x|^?1/N at large |x|. The 1/N expansion is, in fact, a rather good guide to the properties of this model.     

Elliptic genera and quantum field theory

It is shown that in elliptic cohomology — as recently formulated in the mathematical literature — the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the Dirac operator inK-theory. This leads to several insights concerning both elliptic cohomology and string theory. Some of the relevant calculations have been done previously by Schellekens and Warner in a different context.Supported in part by NSF grants PHY 80-19754 and PHY 86-16129     

Gauge theories, vertex models, and quantum groups

It is known that the Jones polynomial of knot theory, and its generalizations, are closely related to the integrable “vertex models” of two-dimensional statistical mechanics, and to quantum groups. In this paper, an attempt is made to show on a priori grounds, starting only from general covariance of three-dimensional Chern-Simons gauge theory and two-dimensional “duality”, why this must be so.     

Pion-nuclear total cross sections near the 32, sol32 resonance

The π⁺ total nuclear cross sections for ⁹Be, ¹²C, ¹⁶O, and ²⁷Al have been measured in the energy range 114 to 210 MeV, while π^? data have been taken at 153 MeV. In the center-of-mass coordinate system the π-nucleus total cross sections reach their maximum value near the π-nucleon $\text{(}\text{3}\text{2}\text{,}\text{3}\text{2}\text{)}$ resonance (T_π^C.M.= 1.52 MeV) where they scale as $\text{A}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. The measured total cross sections exhibit considerable broadening with respect to the π-nucleon resonance.     

Novel motifs for magnetic exchange interactions as exemplified in one dimensional magnets based on the FeCl3-α-di-imine system: Nuclear gamma resonance evidence of slow paramagnetic relaxation and 3D-ordering

The extremes of single ion slow paramagnetic relaxation and cooperative 3D-antiferromagnetic order are exemplified in a series of structurally characterized complexes based on the anhydrous FeCl₃-α-di-imine system. “Soft” Cl−Cl contacts, C−C (π-π) and hydrogen bonding interactions are found to lead to surprisingly strong magnetic exchange among five and six coordinate high-spin iron III monomers. The latter form the building blocks of novel “double chain” polymer magnets held together via the preceding “weak” interactions.     

Eigenvalue inequalities for fermions in gauge theories

We show that QCD with a sufficient number of fermions of zero bare mass has physical massless particles. That result also follows from triangle anomalies, so only our method is novel. Our method involves proving special cases of recently conjectured paramagnetic inequalities for fermions. The proofs are simple applications of the Atiyah-Patodi-Singer theorem on spectral flow.Supported in part by the National Science Foundation Grant No. PHY80-19754Supported in part by DOE Grant No. DE-AC02-76ER02220     

Compression induced folding of a sheet: an integrable system

The apparently intractable shape of a fold in a compressed elastic film lying on a fluid substrate is found to have an exact solution. Such systems buckle at a nonzero wave vector set by the bending stiffness of the film and the weight of the substrate fluid. Our solution describes the entire progression from a weakly displaced sinusoidal buckling to a single large fold that contacts itself. The pressure decrease is exactly quadratic in the lateral displacement. We identify a complex wave vector whose magnitude remains invariant with compression.     

Microscopic folds and macroscopic jerks in compressed lipid monolayers

We report hitherto unrecognized cooperative behavior in the stochastic collapse of certain compressed lipid monolayers implicated in pulmonary function. The cooperativity emerges from a statistical analysis of the collapse events captured using fluorescence microscopy and digital image analysis. The collapse events involve folding of the monolayer on a micron scale, yet each event produces a macroscopic jerk of the layer. The cooperative collapse is striking for its temporal sharpness and large spatial extent.