Computing A Glimpse of Randomness

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000. Full description of programs and proofs will be given elsewhere.Comment: 16 pages; Experimental Mathematics (accepted

Clamping and interlocking effects on IBS block house system in comparison with conventional house system

Industrialised Building System (IBS) is a unique construction technique that has been implemented in many construction fields all around the world. However, its implementation in Malaysia is still slow and not effective. Through the research on IBS, some elements are found to be important and need to be improved in order to produce better quality components. One of the important elements is the design and innovation of IBS components by applying new interlocking configuration between blocks and by using a clamping bolted connection to the system. The main objective of this research is to determine the structural behavior of IBS block works sub-system under push over cyclic loading in comparison with conventional sub-system and to verify that the IBS interlocking geometry sub-system perform better than other sub-systems via laboratory tests. In this research, a block work assembly to form building sub-frame that integrated by two beams, two columns and infill system were built and tested to failure. Two types of IBS block work sub-systems with original geometry and interlocking geometry with scaled of 1:5 were tested with Push Over Cyclic Load Test. In comparison, a control model of Conventional Sub-System was also tested and analysed using the same methods. The results showed that the IBS geometry model with interlocking configuration performed better in terms of stiffness, ductility and flexibility of the models. The IBS original geometry model is ductile but lack structural stiffness while the conventional model is stiff but not ductile

Applicability of the High Field Model: A Preliminary Numerical Study

In a companion presentation, we have discussed the theory of a mesoscopic/ macroscopic model, which can be viewed as an augmented drift-diffusion model. Here, we describe how that model is used. The device we consider for this presentation is the one dimensional GaAs n+−n−n+ structure of length 0.8μm. First, a full Hydro- Dynamic (HD) model, proven reliable when compared with Monte Carlo simulations, is used to simulate the device via the ENO finite difference method. As applied to the full device, the new model is not necessarily superior to traditional Drift-Diffusion (DD). Indeed, when we plot the quantity η= μ0E/kT0/m, where μ0 is the mobility constant and E=−ϕ′ is the electric field, we verify that the high field assumption η › 1, required for the high field model, is satisfied only in an interval given approximately by [0.2, 0.5]. When we run both the DD model and the new high field model in this restricted interval, with boundary conditions of concentration n and potential ϕ provided by the HD results, we demonstrate that the new model outperforms the DD model. This indicates that the high field and DD models should be used only in parts of the device, connected by a transition kinetic regime. This will be a domain decomposition issue involving interface conditions and adequate numerical methods

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general