Log(Rank-1/2): A Simple Way to Improve the OLS Estimation of Tail Exponents

A popular way to estimate a Pareto exponent is to run an OLS regression: log (Rank) = c - blog (Size), and take b as an estimate of the Pareto exponent. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and argue that, if one wants to use an OLS regression, one should use the Rank -1/2, and run log (Rank- 1/2) = c-b log (Size). The shift of 1/2 is optimal, and cancels the bias to a leading order. The standard error on the Pareto exponent is not the OLS standard error, but is asymptotically (2/n)^{1/2}b. To obtain this result, we provide asymptotic expansions for the OLS estimate in such log-log rank-size regression with arbitrary shifts in the ranks. The arguments for the asymptotic expansions rely on strong approximations to martingales with the optimal rate and demonstrate that martingale convergence methods provide a natural and conceptually simple framework for deriving the asymptotics of the tail index estimates using the log-log rank-size regressions.

Inverse Modelling to Obtain Head Movement Controller Signal

Experimentally obtained dynamics of time-optimal, horizontal head rotations have previously been simulated by a sixth order, nonlinear model driven by rectangular control signals. Electromyography (EMG) recordings have spects which differ in detail from the theoretical rectangular pulsed control signal. Control signals for time-optimal as well as sub-optimal horizontal head rotations were obtained by means of an inverse modelling procedures. With experimentally measured dynamical data serving as the input, this procedure inverts the model to produce the neurological control signals driving muscles and plant. The relationships between these controller signals, and EMG records should contribute to the understanding of the neurological control of movements

Convergence of Hill's method for nonselfadjoint operators

By the introduction of a generalized Evans function defined by an appropriate 2- modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrice

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

A simple example is considered of Hill's equation x" + (a^2 + bp(t))x = 0, where the forcing term p, instead of periodic, is quasi periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitrary b, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

Convergence of Hill's Method for Nonselfadjoint Operators

This is the published version, also available here: http://dx.doi.org/10.1137/100809349.By the introduction of a generalized Evans function defined by an appropriate 2-modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices

Geometrically nonlinear analysis of laminated elastic structures

This final technical report contains three parts: Part 1 deals with the 2-D shell theory and its element formulation and applications. Part 2 deals with the 3-D degenerated element. These two parts constitute the two major tasks that were completed under the grant. Another related topic that was initiated during the present investigation is the development of a nonlinear material model. This topic is briefly discussed in Part 3. To make each part self-contained, conclusions and references are included in each part. In the interest of brevity, the discussions presented are relatively brief. The details and additional topics are described in the references cited

Head-on collision of viscous vortex rings

The head-on collision of two identical axisymmetric viscous vortex rings is studied through direct simulations of the incompressible Navier-Stokes equations. The initial vorticity distributions considered are those of Hill's spherical vortex and of rings with circular Gaussian cores, each at Reynolds numbers of about 350 and 1000. The Reynolds number is defined by Gamma/Nu, the ratio of circulation to viscosity. As the vortices approach each other by self-induction, the radii increase by mutual induction, and vorticy cancels through viscous cross-diffusion across the collision plane. Following contact, the vorticity distribution in the core forms a head-tail structure (for the cases considered). The characteristic time of vorticity annihilation is compared with that of a 3D collision experiment and 3D numerical simulations. It is found that the annihilation time is somewhat longer in the axisymmetric case than it is in the symmetry plane of the experiment and 3D numerical simulation. By comparing the annihilatiom time with a viscous timescale and a circulation timescale, it is deduced that both the strain rate due to local effects and to 3D vorticity realignment are important

Application of an Optimized Expert System in Development of Simplified Turbomachinery Modelling

Simulating the ingestion of non-uniform inflow to a fan or compressor requires enormous computational resources if the full details of the flow in the blade rows being studied is to be resolved, since full-wheel unsteady computations are required. A simplified modelling approach exists as an alternative computational option, which is the use of volumetric source terms (body forces) in place of the physical blades. Typically, body force models are manually calibrated with reference to single passage simulation results, and demands significant user experience and expertise. The objective of this thesis is to eliminate the need for experience and expertise during model calibration as much as is practical by employing an automated expert system. The modelling approach employed in this work is the combination of an existing turning force model, and an adaptation of an existing viscous force model. The automated system is implemented into Matlab and makes use of Ansys CFX as the flow solver. User input is required to initialize the system but the procedure then runs through to convergence of the final, calibrated model. Viscous force model coefficients that are traditionally found through an iterative procedure, are instead subjected to a Nelder-Mead optimization process. The machine studied as an example of the application of the automated technique is the NASA stage 67 transonic compressor. At peak efficiency, the isentropic rotor and stage efficiency, and the rotor work coefficient are matched within 1% of their single passage counterparts, a result that is on par with a manually generated body force model. A key finding in this thesis is that the stage efficiency is not the optimal parameter used for calibration of the stator\u27s viscous force model. Despite this finding, the model produced performs sufficiently at off-design conditions not nearing choke. Across the speedline simulated, the model predicts the rotor total temperature ratio, total pressure ratio, and the stage total pressure ratio to within 1.3% of the single passage result. The computational time required for the calibration of the model produced from this work is 23 core-days. Although this computational cost remains relatively high, the removal of nearly all required user experience is achieved

Numerical nonlinear inelastic analysis of stiffened shells of revolution. Volume 1: Theory manual for STARS-2P digital computer program

The theoretical analysis background for the STARS-2P nonlinear inelastic program is discussed. The theory involved is amenable for the analysis of large deflection inelastic behavior in axisymmetric shells of revolution subjected to axisymmetric loadings. The analysis is capable of considering such effects as those involved in nonproportional and cyclic loading conditions. The following are also discussed: orthotropic nonlinear kinematic hardening theory; shell wall cross sections and discrete ring stiffeners; the coupled axisymmetric large deflection elasto-plastic torsion problem; and the provision for the inelastic treatment of smeared stiffeners, isogrid, and waffle wall constructions