The dynamical triangulation model of 3-dimensional Quantum Gravity is defined and studied. We propose two different algorithms for numerical simulations, leading to consistent results. One is the 3-dimensional general...
The dynamical triangulation model of 3-dimensional Quantum Gravity is defined and studied. We propose two different algorithms for numerical simulations, leading to consistent results. One is the 3-dimensional generalization of the bonds flip, another is more sophisticated algorithm, based on Schwinger–Dyson equations. We found such care necessary, because our results appear to be quite unexpected. We simulated up to 60000 tetrahedra and observed none of the feared pathologies like factorial growth of the partition function with volume, or collapse to the branched polymer phase. The volume of the Universe grows exponentially when the bare cosmological constant λ approaches the critical value λ c from above, but the closed Universe exists and has peculiar continuum limit. The Universe compressibility diverges as (λ − λ c ) −2 and the bare Newton constant linearly approaches negative critical value as λ goes to λ c , provided the average curvature is kept at zero. The fractal properties turned out to be the same, as in two dimensions, namely the effective Hausdorff dimension grows logarithmically with the size of the test geodesic sphere.
The response of transport measures (Nusselt number, drag and lift force) for two- and three-dimensional flow past a heated cylinder reaching a chaotic state is investigated numerically using a spectral element discret...
The response of transport measures (Nusselt number, drag and lift force) for two- and three-dimensional flow past a heated cylinder reaching a chaotic state is investigated numerically using a spectral element discretization at a Reynolds number Re = 500. The undisturbed two-dimensional flow remains periodic at this Reynolds number, unless a suitable forcing is applied on the naturally produced system. Three-dimensional simulations establish that three-dimensionality sets in at Re almost-equal-to 200. Successive supercritical states are established through a series of period-doublings, before a chaotic state is reached at a Re almost-equal-to 500. For the two-dimensional forced flow, all transport measures oscillate aperiodically in time and undergo a "crisis," i.e., a sudden and dramatic increase in their amplitude. The corresponding three-dimensional, naturally produced chaotic state corresponds to a less drastic change of the transport quantities with both rms and mean values lower than their two-dimensional counterparts.
It has been observed, in earlier computations of bifurcation diagrams for dissipative partial differential equations, that the use of certain explicit approximate inertial forms can give rise to numerical artifacts su...
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We present a fast numerical method for solving the incompressible Euler's equation in two dimensions for the special case when the flow field can be represented by patches of constant vorticity. The method is an a...
We present a fast numerical method for solving the incompressible Euler's equation in two dimensions for the special case when the flow field can be represented by patches of constant vorticity. The method is an adaptive vortex method in which cells (vortex blobs) of multiple scales are used to represent the patches so that the number of vortex blobs needed to approximate the patches is proportional to the length of the boundary curve of the patch and inversely proportional to the width of the smallest blob (cell) used. Points along the boundaries of the patches are advected according to the velocity obtained from the approximating vortices.
We describe a new approach to the Monte-Carlo simulations of two-dimensional gravity. Standard dynamical triangulation technique was combined with results of direct enumeration of the cubic graphs. As a result we were...
We describe a new approach to the Monte-Carlo simulations of two-dimensional gravity. Standard dynamical triangulation technique was combined with results of direct enumeration of the cubic graphs. As a result we were able to build large (128K vertices) statistically independent random graphs directly. The quantitative correspondence between our results and those obtained by standard methods has been observed. The algorithm proved to be so efficient that we were able to conduct all the simulations, which usually require the most powerful computers, on an Iris workstation. An opportunity to generate large random graphs allowed us to observe that the internal geometry of random surfaces is more complicated than simple fractals. External geometry also proved to be rather peculiar.
We report first results of a large-scale simulation of two-dimensional quantum gravity using the dynamical triangulation model for systems of up to sixteen thousand triangles. Our results for the internal geometry sho...
We report first results of a large-scale simulation of two-dimensional quantum gravity using the dynamical triangulation model for systems of up to sixteen thousand triangles. Our results for the internal geometry show an unexpectedly complicated behavior of the internal volume as function of the internal radius. A simple fractal characterization is inadequate to describe the geometry of the states in the system.
A three‐dimensional computational simulator of nonplanar substrates coated with positive photoresists is presented. The model includes four major steps: projection printing, exposure, post‐exposure baking (PEB), and...
A three‐dimensional computational simulator of nonplanar substrates coated with positive photoresists is presented. The model includes four major steps: projection printing, exposure, post‐exposure baking (PEB), and dissolution. Projection printing is based on Hopkins’ classical work. The exposure model employs the full nonlinear wave equation coupled with the photoactive compound (PAC) bleaching rate equation. These equations are solved using a spectral element iterative scheme. The PEB is treated as a material diffusion equation employing ideas introduced by Mack and the dissolution algorithm is our LEAD (least action dissolution) algorithm modified for nonplanar substrates. Several realistic examples are presented displaying final profiles at various dissolution times.
S. Kida, M. Takaoka, F. Hussain; Corrigendum:‘‘Reconnection of two vortex rings’’ [Phys. Fluids A 1, 630 (1989)]Comments, Physics of Fluids A: Fluid Dynamics, V
S. Kida, M. Takaoka, F. Hussain; Corrigendum:‘‘Reconnection of two vortex rings’’ [Phys. Fluids A 1, 630 (1989)]Comments, Physics of Fluids A: Fluid Dynamics, V
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