Theory of optimum shapes in free-surface flows. Part 2. Minimum drag profiles in infinite cavity flow

The problem considered here is to determine the shape of a symmetric two-dimensional plate so that the drag of this plate in infinite cavity flow is a minimum. With the flow assumed steady and irrotational, and the effects due to gravity ignored, the drag of the plate is minimized under the constraints that the frontal width and wetted arc-length of the plate are fixed. The extremization process yields, by analogy with the classical Euler differential equation, a pair of coupled nonlinear singular integral equations. Although analytical and numerical attempts to solve these equations prove to be unsuccessful, it is shown that the optimal plate shapes must have blunt noses. This problem is next formulated by a method using finite Fourier series expansions, and optimal shapes are obtained for various ratios of plate arc-length to plate width

Unsteady, Free Surface Flows; Solutions Employing the Lagrangian Description of the Motion

Numerical techniques for the solution of unsteady free surface flows are briefly reviewed and consideration is given to the feasibility of methods involving parametric planes where the position and shape of the free surface are known in advance. A method for inviscid flows which uses the Lagrangian description of the motion is developed. This exploits the flexibility in the choice of Lagrangian reference coordinates and is readily adapted to include terms due to inhomogeneity of the fluid. Numerical results are compared in two cases of irrotational flow of a homogeneous fluid for which Lagrangian linearized solutions can be constructed. Some examples of wave run-up on a beach and a shelf are then computed

Suppression of weak-localization (and enhancement of noise) by tunnelling in semiclassical chaotic transport

We add simple tunnelling effects and ray-splitting into the recent trajectory-based semiclassical theory of quantum chaotic transport. We use this to derive the weak-localization correction to conductance and the shot-noise for a quantum chaotic cavity (billiard) coupled to $n$ leads via tunnel-barriers. We derive results for arbitrary tunnelling rates and arbitrary (positive) Ehrenfest time, $\tau_{\rm E}$. For all Ehrenfest times, we show that the shot-noise is enhanced by the tunnelling, while the weak-localization is suppressed. In the opaque barrier limit (small tunnelling rates with large lead widths, such that Drude conductance remains finite), the weak-localization goes to zero linearly with the tunnelling rate, while the Fano factor of the shot-noise remains finite but becomes independent of the Ehrenfest time. The crossover from RMT behaviour ($\tau_{\rm E}=0$) to classical behaviour ($\tau_{\rm E}=\infty$) goes exponentially with the ratio of the Ehrenfest time to the paired-paths survival time. The paired-paths survival time varies between the dwell time (in the transparent barrier limit) and half the dwell time (in the opaque barrier limit). Finally our method enables us to see the physical origin of the suppression of weak-localization; it is due to the fact that tunnel-barriers ``smear'' the coherent-backscattering peak over reflection and transmission modes.Comment: 20 pages (version3: fixed error in sect. VC - results unchanged) - Contents: Tunnelling in semiclassics (3pages), Weak-localization (5pages), Shot-noise (5pages

Automatic Debiased Machine Learning of Causal and Structural Effects

Many causal and structural effects depend on regressions. Examples include average treatment effects, policy effects, average derivatives, regression decompositions, economic average equivalent variation, and parameters of economic structural models. The regressions may be high dimensional. Plugging machine learners into identifying equations can lead to poor inference due to bias and/or model selection. This paper gives automatic debiasing for estimating equations and valid asymptotic inference for the estimators of effects of interest. The debiasing is automatic in that its construction uses the identifying equations without the full form of the bias correction and is performed by machine learning. Novel results include convergence rates for Lasso and Dantzig learners of the bias correction, primitive conditions for asymptotic inference for important examples, and general conditions for GMM. A variety of regression learners and identifying equations are covered. Automatic debiased machine learning (Auto-DML) is applied to estimating the average treatment effect on the treated for the NSW job training data and to estimating demand elasticities from Nielsen scanner data while allowing preferences to be correlated with prices and income

Shot noise in semiclassical chaotic cavities

We construct a trajectory-based semiclassical theory of shot noise in clean chaotic cavities. In the universal regime of vanishing Ehrenfest time \tE, we reproduce the random matrix theory result, and show that the Fano factor is exponentially suppressed as \tE increases. We demonstrate how our theory preserves the unitarity of the scattering matrix even in the regime of finite \tE. We discuss the range of validity of our semiclassical approach and point out subtleties relevant to the recent semiclassical treatment of shot noise in the universal regime by Braun et al. [cond-mat/0511292].Comment: Final version, to appear in Physical Review Letter

Theory of optimum shapes in free-surface flows. Part 1. Optimum profile of sprayless planing surface

This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory

Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity

This paper investigates identification and inference in a nonparametric structural model with instrumental variables and non-additive errors. We allow for non-additive errors because the unobserved heterogeneity in marginal returns that often motivates concerns about endogeneity of choices requires objective functions that are non-additive in observed and unobserved components. We formulate several independence and monotonicity conditions that are sufficient for identification of a number of objects of interest, including the average conditional response, the average structural function, as well as the full structural response function. For inference we propose a two-step series estimator. The first step consists of estimating the conditional distribution of the endogenous regressor given the instrument. In the second step the estimated conditional distribution function is used as a regressor in a nonlinear control function approach. We establish rates of convergence, asymptotic normality, and give a consistent asymptotic variance estimator.