The Laplacian and Dirac operators on critical planar graphs

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and Dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of the Dirac operator and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete holomorphic functions which, via convolutions, gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs. Oblatum 6-III-2002 & 12-VI-2002?Published online: 6 August 2002     

Fibrin proliferation at model surfaces: influence of surface properties

Fibrin proliferation from both human fibrinogen solutions and platelet-poor plasma was studied quantitatively as a function of substrate surface properties. A quartz crystal microbalance was used to monitor both protein adsorption and fibrin proliferation in real time at hydrophobic, hydrophilic, positively charged, and negatively charged surfaces. Scanning electron microscopy was used to characterize the morphology of the polymerized fibrin layers. The observed changes in mass indicate that fibrinogen adsorption occurs rapidly and mediates subsequent fibrin proliferation. Notably, substrate surface properties significantly affect the ability of adsorbed fibrinogen to promote fibrin proliferation.     

Rigidity of planar tilings

Summary Aperturbation of a tiling of a region inR ⁿ is a set of isometries, one applied to each tile, so that the images of the tiles tile the same region.We show that a locally finite tiling of an open region inR ² with tiles which are closures of their interiors isrigid in the following sense: any sufficiently small perturbation of the tiling must have only earthquake-type discontinuities, that is, the discontinuity set consists of straight lines and arcs of circles, and the perturbation near such a curve shifts points along the direction of that curve.We give an example to show that this type of rigidity does not hold inR ⁿ , forn>2.Using rigidity in the plane we show that any tiling problem with a finite number of tile shapes (which are topological disks) is equivalent to a polygonal tiling problem, i.e. there is a set of polygonal shapes with equivalent tiling combinatorics.Oblatum 19-III-1991     

Scale size of magnetic turbulence in tokamaks probed with 30-MeV electrons

Measurements of synchrotron radiation emitted by 30-MeV runaway electrons in the TEXTOR-94 tokamak show that the runaway population decays after switching on neutral beam injection (NBI). The decay starts only with a significant delay, which decreases with increasing NBI heating power. This delay provides direct evidence of the energy dependence of runaway confinement, which is expected if magnetic modes govern the loss of runaways. Application of the theory by Mynick and Strachan [Phys. Fluids 24, 695 (1981)] yields estimates for the "mode width" (delta) of magnetic perturbations: delta<0.5 cm in Ohmic discharges, increasing to delta = 4.4 cm for 0. 6 MW NBI.     

Stable gravity wave of arbitrary amplitude in finite depth

A combination and modification of two existing methods, which involves balancing static and dynamic pressure differences between points along the surface and conserving mass through cross sections below the surface in the reference frame moving with the phase velocity, is applied to surface gravity waves of arbitrary amplitude in water of finite depth. For a given still water depth and wave height the method determines in closed form the phase velocity, wavelength, and wave profile of the stable wave. The main assumption is that the horizontal component of the fluid velocity be independent of depth. The motion is not assumed to be irrotational. The wavelength of the stable wave is found to be about 3.6 times the still water depth for infinitesimal amplitude, and at finite amplitude the wavelength decreases as the amplitude increases. Therefore, shallow water waves are concluded to be unstable even at infinitesimal amplitude, for which the assumption is accurate. Previously it has been argued that only at finite amplitude will shallow water waves change form as they propagate. The wave profile is found to be sinusoidal for infinitesimal amplitude and to be asymmetric at finite amplitude, the crests being higher and narrower and the troughs shallower and broader. These results are consistent with well-known theoretical work and laboratory measurements.     

Projecting the one-dimensional Sierpinski gasket

LetS⊂ℝ² be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1. This work was partially completed while the author was at the Institut Fourier, Grenoble, France.     

Experimental comparison of Jψ production by π±, K±, p and p̄ beams at 39.5 GeV/c

Measurements have been made of relative production cross sections of the $\text{J}\text{ψ}$ by π^±, K^±, p and p? at 39.5 GeV/c incident on copper. $\text{J}\text{ψ}$ production rates from π^?, K^? and p? are similar. The $\text{J}\text{ψ}$ relative particle/anti-particle production cross sections for x>0 are $\text{σ(π}{}^{\mathrm{+}}\text{)}\text{σ(π}{}^{\mathrm{?}}\text{)=(0.87±0.14)}$, σ(K⁺)/σ(K^?)=(0.85±0.5) and $\text{σ(}\text{p}\text{)}\text{σ(}\text{p}\text{?}\text{)=(0.15 ±0.08)}$. The small p/p? cross section ratio disagrees with models of J/ψ production by gluon amalgamation.     

Electron impact induced fragmentations and rearrangements of some phenyl substituted phosphine oxides

Electron impact induced fragmentations and rearrangements of 1-phenyl-2,2,3-trimethylphosphetane 1-oxide, 1-phenylphospholane 1-oxide and 1-phenylphosphorinane 1-oxide were investigated. A process common to all of these phosphine oxides was electron impact induced formation of new carbon-carbon bonds involving the phenyl rings. Evidence for such rearrangements included observation of hydrocarbon fragment ions containing seven or more carbons.