Projective duals to algebraic and tropical hypersurfaces

We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in $\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in terms of $\text{Trop}(X)$, as long as $\text{Trop}(X)$ is smooth and satisfies a mild genericity condition. As a consequence, when $X$ is a curve we describe the transformation of Newton polygons under projective duality, and recover classical formulas for the degree of a dual plane curve. For higher dimensional hypersurfaces $X$, we give a partial description of $\text{Trop}(X^*)$.Comment: 47 pages, 13 figures; v2 minor revisions; accepted to PLM

Unchanging E-Discovery in the Patent Courts

This Article analyzes the Federal Circuit’s Model Order Regarding E-Discovery in Patent Cases (the “Model Order”). The Article briefly describes the purpose behind the Model Order, describes its key provisions, analyzes the Model Order to identify some areas of continuing concern, and defines predictive coding to examine the impact, or lack thereof, on the Model Order. The Author concludes that, while it is beyond refute that the Model Order is an appropriate step toward controlling and managing e-discovery, the Model Order is only the first step. In this regard, several problems, as set forth below, can potentially arise when counsel or the courts use the Model Order. It is hoped that this Article will encourage judges, litigants, and other interested parties to continue trying to solve the continuously troubling aspects of e-discovery and e-discovery abuse

Classical Diffusion of a quantum particle in a noisy environment

We study the spreading of a quantum-mechanical wavepacket in a one-dimensional tight-binding model with a noisy potential, and analyze the emergence of classical diffusion from the quantum dynamics due to decoherence. We consider a finite correlation time of the noisy environment, and treat the system by utilizing the separation of fast (dephasing) and slow (diffusion) processes. We show that classical diffusive behavior emerges at long times, and we calculate analytically the dependence of the classical diffusion coefficient on the noise magnitude and correlation time. This method provides a general solution to this problem for arbitrary conditions of the noisy environment. The results are relevant to a large variety of physical systems, from electronic transport in solid state physics, to light transmission in optical devices, diffusion of excitons, and quantum computation

Multi-View Polarimetric Scattering Cloud Tomography and Retrieval of Droplet Size

Tomography aims to recover a three-dimensional (3D) density map of a medium or an object. In medical imaging, it is extensively used for diagnostics via X-ray computed tomography (CT). We define and derive a tomography of cloud droplet distributions via passive remote sensing. We use multi-view polarimetric images to fit a 3D polarized radiative transfer (RT) forward model. Our motivation is 3D volumetric probing of vertically-developed convectively-driven clouds that are ill-served by current methods in operational passive remote sensing. Current techniques are based on strictly 1D RT modeling and applied to a single cloudy pixel, where cloud geometry defaults to that of a plane-parallel slab. Incident unpolarized sunlight, once scattered by cloud-droplets, changes its polarization state according to droplet size. Therefore, polarimetric measurements in the rainbow and glory angular regions can be used to infer the droplet size distribution. This work defines and derives a framework for a full 3D tomography of cloud droplets for both their mass concentration in space and their distribution across a range of sizes. This 3D retrieval of key microphysical properties is made tractable by our novel approach that involves a restructuring and differentiation of an open-source polarized 3D RT code to accommodate a special two-step optimization technique. Physically-realistic synthetic clouds are used to demonstrate the methodology with rigorous uncertainty quantification

Realization of quantum walks with negligible decoherence in waveguide lattices

Quantum random walks are the quantum counterpart of classical random walks, and were recently studied in the context of quantum computation. Physical implementations of quantum walks have only been made in very small scale systems severely limited by decoherence. Here we show that the propagation of photons in waveguide lattices, which have been studied extensively in recent years, are essentially an implementation of quantum walks. Since waveguide lattices are easily constructed at large scales and display negligible decoherence, they can serve as an ideal and versatile experimental playground for the study of quantum walks and quantum algorithms. We experimentally observe quantum walks in large systems (similar to 100 sites) and confirm quantum walks effects which were studied theoretically, including ballistic propagation, disorder, and boundary related effects

The interaction between colloids in polar mixtures above Tc

We calculate the interaction potential between two colloids immersed in an aqueous mixture containing salt near or above the critical temperature. We find an attractive interaction far from the coexistence curve due to the combination of preferential solvent adsorption at the colloids' surface and preferential ion solvation. We show that the ion-specific interaction strongly depends on the amount of salt added as well as on the mixture composition. Our results are in accord with recent experiments. For a highly antagonistic salt of hydrophilic anions and hydrophobic cations, a repulsive interaction at an intermediate inter-colloid distance is predicted even though both the electrostatic and adsorption forces alone are attractive.Comment: 9 pages, 6 figure

The Effect of Age, Sex, Area Deprivation, and Living Arrangements on Total Knee Replacement Outcomes: A Study Involving the United Kingdom National Joint Registry Dataset

Background: Total knee replacement (TKR) is a common procedure for the treatment of osteoarthritis that provides a substantial reduction of knee pain and improved function in most patients. We investigated whether sociodemographic factors could explain variations in the benefit resulting from TKR. Methods: Data were collected from 3 sources: the National Joint Registry for England, Wales, Northern Ireland, and the Isle of Man; National Health Service (NHS) England Patient Reported Outcome Measures; and Hospital Episode Statistics. These 3 sources were linked for analysis. Pain and function of the knee were measured with use of the Oxford Knee Score (OKS). The risk factors of interest were age group, sex, deprivation, and social support. The outcomes of interest were sociodemographic differences in preoperative scores, 6-month postoperative scores, and change in scores. Results: Ninety-one thousand nine hundred and thirty-six adults underwent primary TKR for the treatment of osteoarthritis in an NHS England unit from 2009 to 2012. Sixty-six thousand seven hundred and sixty-nine of those patients had complete knee score data and were included in the analyses for the present study. The preoperative knee scores were worst in female patients, younger patients, and patients from deprived areas. At 6 months postoperatively, the mean knee score had improved by 15.2 points. There were small sociodemographic differences in the benefit of surgery, with greater area deprivation (−0.71 per quintile of increase in deprivation; 95% confidence interval [CI], −0.76 to −0.66; p < 0.001) and younger age group (−3.51 for ≤50 years compared with 66 to 75 years; 95% CI, −4.00 to −3.02; p < 0.001) associated with less benefit. Cumulatively, sociodemographic factors explained <1% of the total variability in improvement. Conclusions: Sociodemographic factors have a small influence on the benefit resulting from TKR. However, as they are associated with the clinical threshold at which the procedure is performed, they do affect the eventual outcomes of TKR. Level of Evidence: Prognostic Level IV. See Instructions for Authors for a complete description of evidence